Coding techniques using estimated spectral magnitude and phase derived from MDCT coefficients

ABSTRACT

Estimates of spectral magnitude and phase are obtained by an estimation process using spectral information from analysis filter banks such as the Modified Discrete Cosine Transform. The estimation process may be implemented by convolution-like operations with impulse responses. Portions of the impulse responses may be selected for use in the convolution-like operations to trade off between computational complexity and estimation accuracy. Mathematical derivations of analytical expressions for filter structures and impulse responses are disclosed.

TECHNICAL FIELD

The present invention provides an efficient process for accuratelyestimating spectral magnitude and phase from spectral informationobtained from various types of analysis filter banks including thoseimplemented by Modified Discrete Cosine Transforms and Modified DiscreteSine Transforms. These accurate estimates may be used in various signalprocessing applications such as audio coding and video coding.

In the following discussion more particular mention is made of audiocoding applications using filter banks implemented by a particularModified Discrete Cosine Transform; however, the present invention isalso applicable to other applications and other filter bankimplementations.

BACKGROUND ART

Many coding applications attempt to reduce the amount of informationrequired to adequately represent a source signal. By reducinginformation capacity requirements, a signal representation can betransmitted over channels having lower bandwidth or stored on mediausing less space.

Coding can reduce the information capacity requirements of a sourcesignal by eliminating either redundant components or irrelevantcomponents in the signal. So called perceptual coding methods andsystems often use filter banks to reduce redundancy by decorrelating asource signal using a basis set of spectral components, and reduceirrelevancy by adaptive quantization of the spectral componentsaccording to psycho-perceptual criteria. A coding process that adaptsthe quantizing resolution more coarsely can reduce informationrequirements to a greater extent but it also introduces higher levels ofquantization error or “quantization noise” into the signal. Perceptualcoding systems attempt to control the level of quantization noise sothat the noise is “masked” or rendered imperceptible by other spectralcontent of the signal. These systems typically use perceptual models topredict the levels of quantization noise that can be masked by a givensignal.

In perceptual audio coding systems, for example, quantization noise isoften controlled by adapting quantizing resolutions according topredictions of audibility obtained from perceptual models based onpsychoacoustic studies such as that described in E. Zwicker,Psychoacoustics, 1981. An example of a perceptual model that predictsthe audibility of spectral components in a signal is discussed in M.Schroeder et al.; “Optimizing Digital Speech Coders by ExploitingMasking Properties of the Human Ear,” J. Acoust. Soc. Am., December1979, pp. 1647–1652.

Spectral components that are deemed to be irrelevant because they arepredicted to be imperceptible need not be included in the encodedsignal. Other spectral components that are deemed to be relevant can bequantized using a quantizing resolution that is adapted to be fineenough to ensure the quantization noise is rendered just imperceptibleby other spectral components in the source signal. Accurate predictionsof perceptibility by a perceptual model allow a perceptual coding systemto adapt the quantizing resolution more optimally, resulting in feweraudible artifacts.

A coding system using models known to provide inaccurate predictions ofperceptibility cannot reliably ensure quantization noise is renderedimperceptible unless a finer quantizing resolution is used than wouldotherwise be required if a more accurate prediction was available. Manyperceptual models such as that discussed by Schroeder, et al. are basedon spectral component magnitude; therefore, accurate predictions bythese models depend on accurate measures of spectral componentmagnitude.

Accurate measures of spectral component magnitude also influence theperformance of other types of coding processes in addition toquantization. In two types of coding processes known as spectralregeneration and coupling, an encoder reduces information requirementsof source signals by excluding selected spectral components from anencoded representation of the source signals and a decoder synthesizessubstitutes for the missing spectral components. In spectralregeneration, the encoder generates a representation of a basebandportion of a source signal that excludes other portions of the spectrum.The decoder synthesizes the missing portions of the spectrum using thebaseband portion and side information that conveys some measure ofspectral level for the missing portions, and combines the two portionsto obtain an imperfect replica of the original source signal. Oneexample of an audio coding system that uses spectral regeneration isdescribed in international patent application no. PCT/US03/08895 filedMar. 21, 2003, publication no. WO 03/083034 published Oct. 9, 2003. Incoupling, the encoder generates a composite representation of spectralcomponents for multiple channels of source signals and the decodersynthesizes spectral components for multiple channels using thecomposite representation and side information that conveys some measureof spectral level for each source signal channel. One example of anaudio coding system that uses coupling is described in the AdvancedTelevision Systems Committee (ATSC) A/52A document entitled “Revision Ato Digital Audio Compression (AC-3) Standard” published Aug. 20, 2001.

The performance of these coding systems can be improved if the decoderis able to synthesize spectral components that preserve the magnitudesof the corresponding spectral components in the original source signals.The performance of coupling also can be improved if accurate measures ofphase are available so that distortions caused by coupling out-of-phasesignals can be avoided or compensated.

Unfortunately, some coding systems use particular types of filter banksto derive an expression of spectral components that make it difficult toobtain accurate measures of spectral component magnitude or phase. Twocommon types of coding systems are referred to as subband coding andtransform coding. Filter banks in both subband and transform codingsystems may be implemented by a variety of signal processing techniquesincluding various time-domain to frequency-domain transforms. See J.Tribolet et al., “Frequency Domain Coding of Speech,” IEEE Trans.Acoust., Speech, and Signal Proc., ASSP-27, October, 1979, pp. 512–530.

Some transforms such as the Discrete Fourier Transform (DFT) or itsefficient implementation, the Fast Fourier Transform (FFT), provide aset of spectral components or transform coefficients from which spectralcomponent magnitude and phase can be easily calculated. Spectralcomponents of the DFT, for example, are multidimensional representationsof a source signal. Specifically, the DFT, which may be used in audiocoding and video coding applications, provides a set of complex-valuedcoefficients whose real and imaginary parts may be expressed ascoordinates in a two-dimensional space. The magnitude of each spectralcomponent provided by such a transform can be obtained easily from eachcomponent's coordinates in the multi-dimensional space using well knowncalculations.

Some transforms such as the Discrete Cosine Transform, however, providespectral components that make it difficult to obtain an accurate measureof spectral component magnitude or phase. The spectral components of theDCT, for example, represent the spectral component of a source signal inonly a subspace of the multidimensional space required to accuratelyconvey spectral magnitude and phase. In typical audio coding and videocoding applications, for example, a DCT provides a set of real-valuedspectral components or transform coefficients that are expressed in aone dimensional subspace of the two-dimensional real/imaginary spacementioned above. The magnitude of each spectral component provided bytransforms like the DCT cannot be obtained easily from each component'scoordinates in the relevant subspace.

This characteristic of the DCT is shared by a particular ModifiedDiscrete Cosine Transform (MDCT), which is described in J. Princen etal., “Subband/Transform Coding Using Filter Bank Designs Based on TimeDomain Aliasing Cancellation,” ICASSP 1987 Conf. Proc., May 1987, pp.2161–64. The MDCT and its complementary Inverse Modified Discrete CosineTransform (IMDCT) have gained widespread usage in many coding systemsbecause they permit implementation of a critically sampledanalysis/synthesis filter bank system that provides for perfectreconstruction of overlapping segments of a source signal. Perfectreconstruction refers to the property of an analysis/synthesis filterbank pair to reconstruct perfectly a source signal in the absence oferrors caused by finite precision arithmetic. Critical sampling refersto the property of an analysis filter bank to generate a number ofspectral components that is no greater than the number of samples usedto convey the source signal. These properties are very attractive inmany coding applications because critical sampling reduces the number ofspectral components that must be encoded and conveyed in an encodedsignal.

The concept of critical sampling deserves some comment. Although the DFTor the DCT, for example, generate one spectral component for each samplein a source signal segment, DFT and DCT analysis/synthesis systems inmany coding applications do not provide critical sampling because theanalysis transform is applied to a sequence of overlapping signalsegments. The overlap allows use of non-rectangular shaped windowfunctions that improve analysis filter bank frequency responsecharacteristics and eliminate blocking artifacts; however, the overlapalso prevents perfect reconstruction with critical sampling because theanalysis filter bank must generate more coefficient values than thenumber of source signal samples. This loss of critical samplingincreases the information requirements of the encoded signal.

As mentioned above, filter banks implemented by the MDCT and IMDCT areattractive in many coding systems because they provide perfectreconstruction of overlapping segments of a source signal withcritically sampling. Unfortunately, these filter banks are similar tothe DCT in that the spectral components of the MDCT represent thespectral component of a source signal in only a subspace of themultidimensional space required to accurately convey spectral magnitudeand phase. Accurate measures of spectral magnitude or phase cannot beobtained easily from the spectral components or transform coefficientsgenerated by the MDCT; therefore, the coding performance of many systemsthat use the MDCT filter bank is suboptimal because the predictionaccuracy of perceptual models is degraded and the preservation ofspectral component magnitudes by synthesizing processes is impaired.

Prior attempts to avoid this deficiency of various filter banks like theMDCT and DCT filter banks have not been satisfactory for a variety ofreasons. One technique is disclosed in “ISO/IEC 11172–3: 1993 (E) Codingof Moving Pictures and Associated Audio for Digital Storage Media at Upto About 1.5 Mbit/s,” ISO/IEC JTC1/SC29/WG11, Part III Audio. Accordingto this technique, a set of filter banks including several MDCT-basedfilter banks is used to generate spectral components for encoding and anadditional FFT-based filter bank is used to derive accurate measures ofspectral component magnitude. This technique is not attractive for atleast two reasons: (1) considerable computational resources are requiredin the encoder to implement the additional FFT filter bank needed toderive the measures of magnitude, and (2) the processing to obtainaccurate measures of magnitude are performed in the encoder; thereforeadditional bandwidth is required by the encoded signal to convey thesemeasures of spectral component magnitude to the decoder.

Another technique avoids incurring any additional bandwidth required toconvey measures of spectral component magnitude by calculating thesemeasures in the decoder. This is done by applying a synthesis filterbank to the decoded spectral components to recover a replica of thesource signal, applying an analysis filter bank to the recovered signalto obtain a second set of spectral components in quadrature with thedecoded spectral components, and calculating spectral componentmagnitude from the two sets of spectral components. This technique alsois not attractive because considerable computational resources arerequired in the decoder to implement the analysis filter bank needed toobtain the second set of spectral components.

Yet another technique, described in S. Merdjani et al., “DirectEstimation of Frequency From MCT-Encoded Files,” Proc. of the 6th Int.Conf. on Digital Audio Effects (DAFx-03), London, September 2003,estimates the frequency, magnitude and phase of a sinusoidal sourcesignal from a “regularized spectrum” derived from MDCT coefficients.This technique overcomes the disadvantages mentioned above but it alsois not satisfactory for typical coding applications because it isapplicable only for a very simple source signal that has only onesinusoid.

Another technique, which is disclosed in U.S. patent application Ser.No. 09/948,053, publication number U.S. 2003/0093282 A1 published May15, 2003, is able to derive DFT coefficients from MDCT coefficients;however, the disclosed technique does not obtain measures of magnitudeor phase for spectral components represented by the MDCT coefficientsthemselves. Furthermore, the disclosed technique does not use measuresof magnitude or phase to adapt processes for encoding or decodinginformation that represents the MDCT coefficients.

What is needed is a technique that provides accurate estimates ofmagnitude or phase from spectral components generated by analysis filterbanks such as the MDCT that also avoids or overcomes deficiencies ofknown techniques.

DISCLOSURE OF INVENTION

The present invention overcomes the deficiencies of the prior art byreceiving first spectral components that were generated by applicationof an analysis filterbank to a source signal conveying content intendedfor human perception, deriving one or more first intermediate componentsfrom at least some of the first spectral components, forming acombination of the one or more first intermediate components accordingto at least a portion of one or more impulse responses to obtain one ormore second intermediate components, deriving second spectral componentsfrom the one or more second intermediate components, obtaining estimatedmeasures of magnitude or phase using the first spectral components andthe second spectral components, and applying an adaptive process to thefirst spectral components to generate processed information. Theadaptive process adapts in response to the estimated measures ofmagnitude or phase.

The various features of the present invention and its preferredembodiments may be better understood by referring to the followingdiscussion and the accompanying drawings in which like referencenumerals refer to like elements in the several figures. The contents ofthe following discussion and the drawings are set forth as examples onlyand should not be understood to represent limitations upon the scope ofthe present invention.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic block diagram of a transmitter used in a codingsystem.

FIG. 2 is a schematic block diagram of a receiver used in a codingsystem.

FIG. 3 is a schematic block diagram of a device that obtains measures ofspectral component magnitude or phase according to various aspects ofthe present invention.

FIG. 4 is a schematic block diagram of a transmitter that incorporatesvarious aspects of the present invention.

FIG. 5 is a schematic block diagram of a receiver that incorporatesvarious aspects of the present invention.

FIGS. 6–8 are graphical illustrations of impulse responses that may beused with exemplary implementations of the present invention.

FIG. 9 is a schematic block diagram of a device that may be used toimplement various aspects of the present invention.

MODES FOR CARRYING OUT THE INVENTION A. Introduction

The present invention allows accurate measures of magnitude or phase tobe otained from spectral components generated by analysis filter bankssuch as the Modified Discrete Cosine Transform (MDCT) mentioned above.Various aspects of the present invention may be used in a number ofapplications including audio and video coding. FIGS. 1 and 2 illustrateschematic block diagrams of a transmitter and receiver, respectively, ina coding system that may incorporate various aspects of the presentinvention. Features of the illustrated transmitter and receiver arediscussed briefly in the following sections. Following this discussion,features of some analysis and synthesis filter banks that are pertinentto calculating measures of magnitude and phase are discussed.

1. Transmitter

The transmitter illustrated in FIG. 1 applies the analysis filter bank 3to a source signal received from the path 1 to generate spectralcomponents that represent the spectral content of the source signal,applies the encoder 5 to the spectral components to generate encodedinformation, and applies the formatter 8 to the encoded information togenerate an output signal suitable for transmission along the path 9.The output signal may be delivered immediately to a companion receiveror recorded for subsequent delivery. The analysis filter bank 3 may beimplemented in variety of ways including infinite impulse response (IIR)filters, finite impulse response (FIR) filters, lattice filters andwavelet transforms.

Aspects of the present invention are described below with reference toimplementations closely related to the MDCT, however, the presentinvention is not limited to these particular implementations.

In this disclosure, terms like “encoder” and “encoding” are not intendedto imply any particular type of information processing. For example,encoding is often used to reduce information capacity requirements;however, these terms in this disclosure do not necessarily refer to thistype of processing. The encoder 5 may perform essentially any type ofprocessing that is desired. In one implementation, encoded informationis generated by quantizing spectral components according to a perceptualmodel. In another implementation, the encoder 5 applies a couplingprocess to multiple channels of spectral components to generate acomposite representation. In yet another implementation, spectralcomponents for a portion of a signal bandwidth are discarded and anestimate of the spectral envelope of the discarded portion is includedin the encoded information. No particular type of encoding is importantto the present invention.

2. Receiver

The receiver illustrated in FIG. 2 applies the deformatter 23 to aninput signal received from the path 21 to obtain encoded information,applies the decoder 25 to the encoded information to obtain spectralcomponents representing the spectral content of a source signal, andapplies the synthesis filter bank 27 to the spectral components togenerate an output signal that is a replica of the source signal but maynot be an exact replica. The synthesis filter bank 27 may be implementedin a variety of ways that are complementary to the implementation of theanalysis filter bank 3.

In this disclosure, terms like “decoder” and “decoding” are not intendedto imply any particular type of information processing. The decoder 25may perform essentially any type of processing that is needed ordesired. In one implementation that is inverse to an encoding processdescribed above, quantized spectral components are decoded intodequantized spectral components. In another implementation, multiplechannels of spectral components are synthesized from a compositerepresentation of spectral components. In yet another implementation,the decoder 25 synthesizes missing portions of a signal bandwidth fromspectral envelope information. No particular type of decoding isimportant to the present invention.

3. Measures of Magnitude and Phase

In one implementation by an Odd Discrete Fourier Transform (ODFT), theanalysis filter bank 3 generates complex-valued coefficients or“spectral components” with real and imaginary parts that may beexpressed in a two-dimensional space. This transform may be expressedas:

$\begin{matrix}{{X_{ODFT}(k)} = {\sum\limits_{n = 0}^{N - 1}\;{{x(n)} \cdot {\exp\left\lbrack {{- j}\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {n + n_{0}} \right)} \right\rbrack}}}} & (1)\end{matrix}$which may be separated into real and imaginary partsX_(ODFT)(k)=Re[X_(ODFT)(k)]+j·Im[X_(ODFT)(k)]  (2)and rewritten as

$\begin{matrix}{{X_{ODFT}(k)} = {{\sum\limits_{n = 0}^{N - 1}\;{{x(n)} \cdot {\cos\left\lbrack {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {n + n_{0}} \right)} \right\rbrack}}} - {j \cdot {\sum\limits_{n = 0}^{N - 1}\;{{x(n)} \cdot {\sin\left\lbrack {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {n + n_{0}} \right)} \right\rbrack}}}}}} & (3)\end{matrix}$where X_(ODFT)(k)=ODFT coefficient for spectral component k,

-   -   x(n)=source signal amplitude at time n;    -   Re[X]=real part of X; and    -   Im[X]=imaginary part of X.

The magnitude and phase of each spectral component k may be calculatedas follows:

$\begin{matrix}\begin{matrix}{{{Mag}\left\lbrack {X_{ODFT}(k)} \right\rbrack} = {{X_{ODFT}(k)}}} \\{= \sqrt{{{Re}\left\lbrack {X_{ODFT}(k)} \right\rbrack}^{2} + {{Im}\left\lbrack {X_{ODFT}(k)} \right\rbrack}^{2}}}\end{matrix} & (4) \\{{{Phs}\left\lbrack {X_{ODFT}(k)} \right\rbrack} = {\arctan\left\lbrack \frac{{Im}\left\lbrack {X_{ODFT}(k)} \right\rbrack}{{Re}\left\lbrack {X_{ODFT}(k)} \right\rbrack} \right\rbrack}} & (5)\end{matrix}$where Mag[X]=magnitude of X; and

-   -   Phs[X]=phase of X.

Many coding applications implement the analysis filter bank 3 byapplying the Modified Discrete Cosine Transform (MDCT) discussed aboveto overlapping segments of the source signal that are modulated by ananalysis window function. This transform may be expressed as:

$\begin{matrix}{{X_{MDCT}(k)} = {\sum\limits_{n = 0}^{N - 1}\;{{x(n)} \cdot {\cos\left\lbrack {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {n + n_{0}} \right)} \right\rbrack}}}} & (6)\end{matrix}$where X_(MDCT)(k)=MDCT coefficient for spectral component k. It may beseen that the spectral components that are generated by the MDCT areequivalent to the real part of the ODFT coefficients.X_(MDCT)(k)=Re[X_(ODFT)(k)]  (7)

A particular Modified Discrete Sine Transform (MDST) that generatescoefficients representing spectral components in quadrature with thespectral components represented by coefficients of the MDCT may beexpressed as:

$\begin{matrix}{{X_{MDST}(k)} = {\sum\limits_{n = 0}^{N - 1}\;{{x(n)} \cdot {\sin\left\lbrack {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {n + n_{0}} \right)} \right\rbrack}}}} & (8)\end{matrix}$where X_(MDST)(k)=MDST coefficient for spectral component k. It may beseen that the spectral components that are generated by the MDST areequivalent to the negative imaginary part of the ODFT coefficients.X_(MDST)(k)=−Im[X_(ODFT)(k)]  (9)

Accurate measures of magnitude and phase cannot be calculated directlyfrom MDCT coefficients but they can be calculated directly from acombination of MDCT and MDST coefficients, which can be seen bysubstituting equations 7 and 9 into equations 4 and 5:

$\begin{matrix}{{{Mag}\left\lbrack {X_{ODFT}(k)} \right\rbrack} = \sqrt{{{X_{MDCT}}^{2}(k)} + {{X_{MDST}}^{2}(k)}}} & (10) \\{{{Phs}\left\lbrack {X_{ODFT}(k)} \right\rbrack} = {\arctan\left\lbrack \frac{- {X_{MDST}(k)}}{X_{MDCT}(k)} \right\rbrack}} & (11)\end{matrix}$

The Princen paper mentioned above indicates that a correct use of theMDCT requires the application of an analysis window function thatsatisfies certain design criteria. The expressions of transformequations in this section of the disclosure omit an explicit referenceto any analysis window function, which implies a rectangular analysiswindow function that does not satisfy these criteria. This does notaffect the validity of expressions 10 and 11.

Implementations of the present invention described below obtain measuresof spectral component magnitude and phase from MDCT coefficients andfrom MDST coefficients derived from the MDCT coefficients. Theseimplementations are described below following a discussion of theunderlying mathematical basis.

B. Derivation of Mathematical Framework

This section discusses the derivation of an analytical expression forcalculating exact MDST coefficients from MDCT coefficients. Thisexpression is shown below in equations 41a and 41b. The derivations ofsimpler analytical expressions for two specific window functions arealso discussed. Considerations for practical implementations arepresented following a discussion of the derivations.

One implementation of the present invention discussed below is derivedfrom a process for calculating exact MDST coefficients from MDCTcoefficients. This process is equivalent to another process that appliesan Inverse Modified Discrete Cosine Transform (IMDCT) synthesis filterbank to blocks of MDCT coefficients to generate windowed segments oftime-domain samples, overlap-adds the windowed segments of samples toreconstruct a replica of the original source signal, and applies an MDSTanalysis filter bank to a segment of the recovered signal to generatethe MDST coefficients.

1. Arbitrary Window Function

Exact MDST coefficients cannot be calculated from a single segment ofwindowed samples that is recovered by applying the IMDCT synthesisfilter bank to a single block of MDCT coefficients because the segmentis modulated by an analysis window function and because the recoveredsamples contain time-domain aliasing. The exact MDST coefficients can becomputed only with the additional knowledge of the MDCT coefficients forthe preceding and subsequent segments. For example, in the case wherethe segments overlap one another by one-half the segment length, theeffects of windowing and the time-domain aliasing for a given segment IIcan be canceled by applying the synthesis filter bank and associatedsynthesis window function to three blocks of MDCT coefficientsrepresenting three consecutive overlapping segments of the sourcesignal, denoted as segment I, segment II and segment III. Each segmentoverlaps an adjacent segment by an amount equal to one-half of thesegment length. Windowing effects and time-domain aliasing in the firsthalf of segment II are canceled by an overlap-add with the second halfof segment I, and these effects in the second half of segment II arecanceled by an overlap-add with the first half of segment III.

The expression that calculates MDST coefficients from MDCT coefficientsdepends on the number of segments of the source signal, the overlapstructure and length of these segments, and the choice of the analysisand synthesis window functions. None of these features are important inprinciple to the present invention. For ease of illustration, however,it is assumed in the examples discussed below that the three segmentshave the same length N, which is even, and overlap one another by anamount equal to one-half the segment length, that the analysis andsynthesis window functions are identical to one another, that the samewindow functions are applied to all segments of the source signal, andthat the window functions are such that their overlap-add propertiessatisfy the following criterion, which is required for perfectreconstruction of the source signal as explained in the Princen paper.

${{w(r)}^{2} + {w\left( {r + \frac{N}{2}} \right)}^{2}} = {{1\mspace{14mu}{for}\mspace{14mu} r} \in \left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack}$where w(r)=analysis and synthesis window function; and

-   -   N=length of each source signal segment.    -   The MDCT coefficients X_(i) for the source signal x(n) in each        of the segments i may be expressed as:

$\begin{matrix}{{X_{I}(p)} = {\sum\limits_{n = 0}^{N - 1}\;{{w(n)}{x(n)}{\cos\left( {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {n + n_{0}} \right)} \right)}}}} & (12) \\{{X_{II}(p)} = {\sum\limits_{n = 0}^{N - 1}\;{{w(n)}{x\left( {n + \frac{N}{2}} \right)}{\cos\left( {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {n + n_{0}} \right)} \right)}}}} & (13) \\{{X_{III}(p)} = {\sum\limits_{n = 0}^{N - 1}\;{{w(n)}{x\left( {n + N} \right)}{\cos\left( {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {n + n_{0}} \right)} \right)}}}} & (14)\end{matrix}$

The windowed time-domain samples {circumflex over (x)} that are obtainedfrom an application of the IMDCT synthesis filter bank to each block ofMDCT coefficients may be expressed as:

$\begin{matrix}{{{\hat{x}}_{I}(r)} = {\frac{2{w(r)}}{N}{\sum\limits_{p = 0}^{N - 1}\;{{X_{I}(p)}{\cos\left( {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}}}} & (15) \\{{{\hat{x}}_{II}(r)} = {\frac{2{w(r)}}{N}{\sum\limits_{p = 0}^{N - 1}\;{{X_{II}(p)}{\cos\left( {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}}}} & (16) \\{{{\hat{x}}_{III}(r)} = {\frac{2{w(r)}}{N}{\sum\limits_{p = 0}^{N - 1}\;{{X_{III}(p)}{\cos\left( {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}}}} & (17)\end{matrix}$

Samples s(r) of the source signal for segment II are reconstructed byoverlapping and adding the three windowed segments as described above,thereby removing the time-domain aliasing from the source signal x. Thismay be expressed as:

$\begin{matrix}{{s(r)} = \left\{ \begin{matrix}{{{\hat{x}}_{I}\left( {r + \frac{N}{2}} \right)} + {{\hat{x}}_{II}(r)}} & {{{for}\mspace{14mu} r} \in \left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack} \\{{{\hat{x}}_{II}(r)} + {{\hat{x}}_{III}\left( {r - \frac{N}{2}} \right)}} & {{{for}\mspace{14mu} r} \in \left\lbrack {\frac{N}{2},{N - 1}} \right\rbrack}\end{matrix} \right.} & (18)\end{matrix}$

A block of MDST coefficients S(k) may be calculated for segment II byapplying an MDST analysis filter bank to the time-domain samples in thereconstructed segment II, which may be expressed as:

$\begin{matrix}{{S(k)} = {\sum\limits_{r = 0}^{N - 1}{{w(r)}{s(r)}{\sin\left( {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}}} & (19)\end{matrix}$

Using expression 18 to substitute for s(r), expression 19 can berewritten as:

$\begin{matrix}{{S(k)} = {{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{{w(r)}\left\lbrack {{{\hat{x}}_{I}\left( {r + \frac{N}{2}} \right)} + {{\hat{x}}_{II}(r)}} \right\rbrack}{\sin\left( {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}} + {\sum\limits_{r = \frac{N}{2}}^{N - 1}{{{w(r)}\left\lbrack {{{\hat{x}}_{II}(r)} + {{\hat{x}}_{III}\left( {r - \frac{N}{2}} \right)}} \right\rbrack}{\sin\left( {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}}}} & (20)\end{matrix}$This equation can be rewritten in terms of the MDCT coefficients byusing expressions 15–17 to substitute for the time-domain samples:

$\begin{matrix}{{S(k)} = {\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}\left( {{\frac{w\left( {r + \frac{N}{2}} \right)}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{I}(p)}{\cos\left( {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}{\sin\left( {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}}} + {\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}\left( {{\frac{w(r)}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\cos\left( {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}{\sin\left( {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}}} + {\sum\limits_{r = \frac{N}{2}}^{N - 1}{{w(r)}\left( {{\frac{w(r)}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\cos\left( {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}\sin\left( {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}} + {\quad{\quad{\sum\limits_{r = \frac{N}{2}}^{N - 1}{{w(r)}\left( {\frac{w\left( {r - \frac{N}{2}} \right)}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\cos\left( {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}}} \right){{s{in}}\left( {\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} \right)}}}}}} \right.}}} \right.}}} \right.}}} & (21)\end{matrix}$The remainder of this section of the disclosure shows how this equationcan be simplified as shown below in equations 41a and 41b.

Using the trigonometric identity sin α·cos β=½[sin (α+β)+ sin (α−β)] togather terms and switching the order of summation, expression 21 can berewritten as

$\begin{matrix}\begin{matrix}{{S(k)} = {\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{I}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{{w\left( {r + \frac{N}{2}} \right)} \cdot}}}}}}} \\{{\sin\left\lbrack {{\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} + {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} + {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( \frac{N}{2} \right)}} \right\rbrack} +} \\{\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{I}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{{w\left( {r + \frac{N}{2}} \right)} \cdot}}}}}} \\{{\sin\left\lbrack {{\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} + {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} - {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( \frac{N}{2} \right)}} \right\rbrack} +} \\{{\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k + p + 1} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} +} \\{{\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p + 1} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} +} \\{{\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\sum\limits_{r = \frac{N}{2}}^{N - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k + p + 1} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} +} \\{{\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\sum\limits_{r = \frac{N}{2}}^{N - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} +} \\{\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{III}(p)}{\sum\limits_{r = \frac{N}{2}}^{N - 1}{{w(r)}{{w\left( {r - \frac{N}{2}} \right)} \cdot}}}}}} \\{{\sin\left\lbrack {{\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} + {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} - {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( \frac{N}{2} \right)}} \right\rbrack} +} \\{\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{III}(p)}{\sum\limits_{r = \frac{N}{2}}^{N - 1}{{w(r)}{{w\left( {r - \frac{N}{2}} \right)} \cdot}}}}}} \\{\sin\left\lbrack {{\frac{2\pi}{N}\left( {k + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} + {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( {r + n_{0}} \right)} - {\frac{2\pi}{N}\left( {p + \frac{1}{2}} \right)\left( \frac{N}{2} \right)}} \right\rbrack}\end{matrix} & (22)\end{matrix}$

This expression can be simplified by combining pairs of terms that areequal to each other. The first and second terms are equal to each other.The third and fourth terms are equal to each other. The fifth and sixthterms are equal to each other and the seventh and eighth terms are equalto each other. The equality between the third and fourth terms, forexample, may be shown by proving the following lemma:

$\begin{matrix}{{\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k + p + 1} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} = {\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}}} & (23)\end{matrix}$

This lemma may be proven by rewriting the left-hand and right-hand sidesof equation 23 as functions of p as follows:

$\begin{matrix}{{\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k + p + 1} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} = {\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{F(p)}}}} & \left( {24a} \right) \\{{\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} = {\frac{1}{N}{\sum\limits_{p = 0}^{N - 1}{G(p)}}}} & \left( {24b} \right)\end{matrix}$where

$\begin{matrix}{{F(p)} = {{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k + p + 1} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}} & \left( {25a} \right) \\{{G(p)} = {{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}} & \left( {25b} \right)\end{matrix}$The expression of G as a function of (p) can be rewritten as a functionof (N−1−p) as follows:

$\begin{matrix}{{G\left( {N - 1 - p} \right)} = {{X_{II}\left( {N - 1 - p} \right)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - {\left( {N - 1 - p} \right)\left( {r + n_{0}} \right)}} \right\rbrack} \right.}}}}} & (26)\end{matrix}$

It is known that MDCT coefficients are odd symmetric; therefore,X_(II)(N−1−p)=−X_(II)(p) for

$p \in {\left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack.}$By rewriting (k−(N−1−p)) as (k+1+p)−N, it may be seen that(k−(N−1−p))·(r+n₀)=(k+1+p)·(r+n₀)−N·(r+n₀). These two equalities allowexpression 26 to be rewritten as:

$\begin{matrix}{{G\left( {N - 1 - p} \right)} = {{- {X_{II}(p)}}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {{\frac{2\pi}{N}\left( {k + p + 1} \right)\left( {r + n_{0}} \right)} - {2{\pi\left( {r + n_{0}} \right)}}} \right\rbrack}}}}} & (27)\end{matrix}$Referring to the Princen paper, the value for n₀ is ½(N/2+1), which ismid-way between two integers. Because r is an integer, it can be seenthat the final term 27π(r+n₀) in the summand of expression 27 is equalto an odd integer multiple of π; therefore, expression 27 can berewritten as

$\begin{matrix}\begin{matrix}{{G\left( {N - 1 - p} \right)} = {{+ {X_{II}(p)}}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k + p + 1} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}} \\{= {F(p)}}\end{matrix} & (28)\end{matrix}$which proves the lemma shown in equation 23. The equality between theother pairs of terms in equation 22 can be shown in a similar manner.

By omitting the first, third, fifth and seventh terms in expression 22and doubling the second, fourth, sixth and eighth terms, equation 22 canbe rewritten as follows after simplifying the second and eighth terms:

$\begin{matrix}{{S(k)} = {{\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{I}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w\left( {r + \frac{N}{2}} \right)}{\sin\left\lbrack {{\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} - {\pi p} - \frac{\pi}{2}} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{II}(p)}{\sum\limits_{r = \frac{N}{2}}^{N - 1}{w(r){w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}{{X_{III}(p)}{\sum\limits_{r = \frac{N}{2}}^{N - 1}{{w(r)}{w\left( {r - \frac{N}{2}} \right)}{\sin\left\lbrack {{\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} + {\pi p} + \frac{\pi}{2}} \right\rbrack}}}}}}}} & (29)\end{matrix}$Using the following identities:

$\begin{matrix}{{{\sin\left( {\alpha \pm {\pi\; p}} \right)} = {\left( {- 1} \right)^{p}\sin\;\alpha}}{{\sin\left( {\alpha + \frac{\pi}{2}} \right)} = {{+ \cos}\;\alpha}}{{\sin\left( {\alpha - \frac{\pi}{2}} \right)} = {{- \cos}\;\alpha}}} & (30)\end{matrix}$expression 29 can be rewritten as:

$\begin{matrix}{{S(k)} = {{\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left( {- 1} \right)^{p + 1}{X_{I}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w(r)}{w\left( {r + \frac{N}{2}} \right)}{\cos\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{{X_{II}(p)}{\sum\limits_{r - \frac{N}{2}}^{N - 1}\;{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left( {- 1} \right)^{p}{X_{III}(p)}{\sum\limits_{r = \frac{N}{2}}^{N - 1}\;{{w(r)}{w\left( {r - \frac{N}{2}} \right)}{\cos\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}}}} & (31)\end{matrix}$

The inner summations of the third and fourth terms are changed so thattheir limits of summation are from r=0 to r=(N/2−1) by making thefollowing substitutions:

${\sin\left( {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0} + \frac{N}{2}} \right)} \right)} = \;{\left( {- 1} \right)^{k - p}{\sin\left( {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right)}}$${\cos\left( {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0} + \frac{N}{2}} \right)} \right)} = \;{\left( {- 1} \right)^{k - p}{\cos\left( {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right)}}$This allows equation 31 to be rewritten as

$\begin{matrix}{{S(k)} = {{\frac{2}{N}{\sum\limits_{p - 0}^{N - 1}\;{\left( {- 1} \right)^{p + 1}{X_{I}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w(r)}{w\left( {r + \frac{N}{2}} \right)}{\cos\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w(r)}{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left( {- 1} \right)^{({k - p})}{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w\left( {r + \frac{N}{2}} \right)}{w\left( {r + \frac{N}{2}} \right)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left( {- 1} \right)^{p}\left( {- 1} \right)^{({k - p})}{X_{III}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{w\left( {r + \frac{N}{2}} \right)}{w(r)}{\cos\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}}}} & (32)\end{matrix}$

Equation 32 can be simplified by using the restriction imposed on thewindow function mentioned above that is required for perfectreconstruction of the source signal. This restriction isw(r)²+w(r+N/2)²=1. With this restriction, equation 31 can be simplifiedto

$\begin{matrix}{{S(k)} = {{\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left\lbrack {{\left( {- 1} \right)^{p + 1}{X_{I}(p)}} + {\left( {- 1} \right)^{k}{X_{III}(p)}}} \right\rbrack \cdot {\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w(r)}{w\left( {r + \frac{N}{2}} \right)}{\cos\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p - 0}^{N - 1}\;{\left( {- 1} \right)^{({k - p})}{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\left( {1 - {w^{2}(r)}} \right){\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}}}} & (33)\end{matrix}$Gathering terms, equation 33 can be rewritten as

$\begin{matrix}{{S(k)} = {{\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left\lbrack {{\left( {- 1} \right)^{p + 1}{X_{I}(p)}} + {\left( {- 1} \right)^{k}{X_{III}(p)}}} \right\rbrack \cdot {\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w(r)}{w\left( {r + \frac{N}{2}} \right)}{\cos\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left\lbrack {{X_{II}(p)} - {\left( {- 1} \right)^{({k - p})}{X_{II}(p)}}} \right\rbrack{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w^{2}(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left( {- 1} \right)^{({k - p})}{X_{II}(p)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}}} & (34)\end{matrix}$

Equation 34 can be simplified by recognizing the inner summation of thethird term is equal to zero. This can be shown by proving two lemmas.One lemma postulates the following equality:

$\begin{matrix}{{I_{\alpha,q}(r)} = {{\sum\limits_{r - 0}^{\frac{N}{2} - 1}{\sin\left( {\frac{2\pi}{N}(q)\left( {r + \alpha} \right)} \right)}} = {{\sin\left( {\frac{2\pi\; q\;\alpha}{N} + \frac{\pi\; q}{2} - \frac{\pi\; q}{N}} \right)}\frac{\sin\frac{\pi\; q}{2}}{\sin\frac{\pi\; q}{N}}}}} & (35)\end{matrix}$

This equality may be proven by rewriting the summand into exponentialform, rearranging, simplifying and combining terms as follows:

$\begin{matrix}\begin{matrix}{{I_{\alpha,q}(r)} = {\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\frac{1}{2i}\left\lbrack {{\exp\left( {{+ j}\frac{2\pi\; q}{N}\left( {r + \alpha} \right)} \right)} - {\exp\left( {{- j}\frac{2\pi\; q}{N}\left( {r + \alpha} \right)} \right)}} \right\rbrack}}} \\{= {{\frac{1}{2i}{\exp\left( {{+ j}\frac{2\pi\;{qa}}{N}} \right)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\exp\left( {{+ j}\frac{2\pi\;{qr}}{N}} \right)}}} -}} \\{\frac{1}{2i}{\exp\left( {{- j}\frac{2\pi\;{qa}}{N}} \right)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\exp\left( {{- j}\frac{2\pi\;{qr}}{N}} \right)}}} \\{= {{\frac{1}{2i}{{\exp\left( {{+ j}\frac{2\pi\;{qa}}{N}} \right)}\left\lbrack \frac{1 - {\exp\left( {{+ j}\frac{2\pi\; q}{N}\frac{N}{2}} \right)}}{1 - {\exp\left( {{+ j}\frac{2\pi\; q}{N}} \right.}} \right\rbrack}} -}} \\{\frac{1}{2i}{{\exp\left( {{- j}\frac{2\pi\;{qa}}{N}} \right)}\left\lbrack \frac{1 - {\exp\left( {{- j}\frac{2\pi\; q}{N}\frac{N}{2}} \right)}}{1 - {\exp\left( {{- j}\frac{2\pi\; q}{N}} \right.}} \right\rbrack}} \\{= {{\frac{1}{2i}{\exp\left( {{+ j}\frac{2\pi\;{qa}}{N}} \right)}{\frac{\exp\left( {{+ j}\frac{\pi\; q}{2}} \right)}{\exp\left( {{+ j}\frac{\pi\; q}{N}} \right)}\left\lbrack \frac{{\exp\left( {{- j}\frac{\pi\; q}{2}} \right)} - {\exp\left( {{+ j}\frac{\pi\; q}{2}} \right)}}{{\exp\left( {{- j}\frac{\pi\; q}{2}} \right)} - {\exp\left( {{+ j}\frac{\pi\; q}{N}} \right)}} \right\rbrack}} -}} \\{\frac{1}{2i}{\exp\left( {{- j}\frac{2\pi\;{qa}}{N}} \right)}{\frac{\exp\left( {{- j}\frac{\pi\; q}{2}} \right)}{\exp\left( {{- j}\frac{\pi\; q}{N}} \right)}\left\lbrack \frac{{\exp\left( {{+ j}\frac{\pi\; q}{2}} \right)} - {\exp\left( {{- j}\frac{\pi\; q}{2}} \right)}}{{\exp\left( {{+ j}\frac{\pi\; q}{N}} \right)} - {\exp\left( {{- j}\frac{\pi\; q}{2}} \right)}} \right\rbrack}} \\{= {{\frac{1}{2i}{\exp\left( {{{+ j}\frac{2\pi\;{qa}}{N}} + {j\frac{\pi\; q}{2}} - {j\frac{\pi\; q}{2}}} \right)}\frac{\sin\frac{\pi\; q}{2}}{\sin\frac{\pi\; q}{N}}} -}} \\{\frac{1}{2i}{\exp\left( {{{- j}\frac{2\pi\;{qa}}{N}} - {j\frac{\pi\; q}{2}} + {j\frac{\pi\; q}{N}}} \right)}\frac{\sin\frac{\pi\; q}{2}}{\sin\frac{\pi\; q}{N}}} \\{{I_{\alpha,q}(r)} = {{\sin\left( {\frac{2\pi\;{qa}}{N} + \frac{\pi\; q}{2} - \frac{\pi\; q}{N}} \right)}\frac{\sin\frac{\pi\; q}{2}}{\sin\frac{\pi\; q}{N}}}}\end{matrix} & (36)\end{matrix}$

The other lemma postulates

${\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}} = {{0\mspace{14mu}{for}\mspace{14mu} n_{0}} = {\frac{1}{2}{\left( {\frac{N}{2} + 1} \right).}}}$This may be proven by substituting n₀ for a in expression 35 to obtainthe following:

$\begin{matrix}\begin{matrix}{I_{n_{0,q}} = {{\sin\left( {\frac{2\pi\;{q\left( {\frac{N}{2} + 1} \right)}}{N} + \frac{\pi\; q}{2} - \frac{\pi\; q}{N}} \right)}\frac{\sin\frac{\pi\; q}{2}}{\sin\frac{\pi\; q}{N}}}} \\{= {{\sin\left( {{\frac{\pi\; q}{N}\left( {\frac{N}{2} + 1} \right)} + \frac{\pi\; q}{2} - \frac{\pi\; q}{N}} \right)}\frac{\sin\frac{\pi\; q}{2}}{\sin\frac{\pi\; q}{N}}}} \\{= {{\sin\left( {\frac{\pi\; q}{2} + \frac{\pi\; q}{N} + \frac{\pi\; q}{2} - \frac{\pi\; q}{N}} \right)}\frac{\sin\frac{\pi\; q}{2}}{\sin\frac{\pi\; q}{N}}}} \\{{= {{{\sin\left( {\pi\; q} \right)}\frac{\sin\frac{\pi\; q}{2}}{\sin\frac{\pi\; q}{N}}} = {0\mspace{14mu}{for}\mspace{14mu} q}}},{{an}\mspace{14mu}{{integer}.}}}\end{matrix} & (37)\end{matrix}$By substituting (k−p) for q in expression 35 and using the preceding twolemmas, the inner summation of the third term in equation 34 may beshown to equal zero as follows:

${\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\sin\left\lbrack {\frac{2\pi}{N}{q\left( {r + n_{0}} \right)}} \right\rbrack}} = {{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}} = 0}$${{for}\mspace{14mu} n_{0}} = {\frac{1}{2}{\left( {\frac{N}{2} + 1} \right).}}$

Using this equality, equation 34 may be simplified to the following:

$\begin{matrix}{{S(k)} = {{\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left\lbrack {{\left( {- 1} \right)^{p + 1}{X_{I}(p)}} + {\left( {- 1} \right)^{k}{X_{III}(p)}}} \right\rbrack \cdot {\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w(r)}{w\left( {r + \frac{N}{2}} \right)}{\cos\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p - 0}^{N - 1}\;{\left\lbrack {\left\lbrack {1 - \left( {- 1} \right)^{({k - p})}} \right\rbrack{X_{II}(p)}} \right\rbrack{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w^{2}(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {k - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}}}} & (38)\end{matrix}$

The MDST coefficients S(k) of a real-valued signal are symmetricaccording to the expression S(k)=S(N−1−k), for kε[0, N−1]. Using thisproperty, all even numbered coefficients can be expressed asS(2v)=S(N−1−2v)=S(N−2(v+1)+1), for

$v \in {\left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack.}$Because N and 2(v+1) are both even numbers, the quantity (N−2(v+1)+1) isan odd number. From this, it can be seen the even numbered coefficientscan be expressed in terms of the odd numbered coefficients. Using thisproperty of the coefficients, equation 38 can be rewritten as follows:

$\begin{matrix}{{{S\left( {2v} \right)} = {{\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left\lbrack {{\left( {- 1} \right)^{p + 1}{X_{I}(p)}} + {X_{III}(p)}} \right\rbrack \cdot {\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w(r)}{w\left( {r + \frac{N}{2}} \right)}{\cos\left\lbrack {\frac{2\pi}{N}\left( {{2v} - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left\lbrack {\left( {1 - \left( {- 1} \right)^{- p}} \right){X_{II}(p)}} \right\rbrack{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w^{2}(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {{2v} - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}}}}\mspace{14mu}} & (39) \\{{{{where}\mspace{14mu} k} = {2v}},{v \in \left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack}} & \;\end{matrix}$

The second term in this equation is equal to zero for all even values ofp. The second term needs to be evaluated only for odd values of p, orfor p=2l+1 for

$\begin{matrix}{\mspace{76mu}{l \in {\left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack.}}} & \; \\{{{S\left( {2v} \right)} = {{\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{\left\lbrack {{\left( {- 1} \right)^{p + 1}{X_{I}(p)}} + {X_{III}(p)}} \right\rbrack \cdot {\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w(r)}{w\left( {r + \frac{N}{2}} \right)}{\cos\left\lbrack {\frac{2\pi}{N}\left( {{2v} - p} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}} + {\frac{4}{N}{\sum\limits_{l = 0}^{\frac{N}{2} - 1}\;{{X_{II}\left( {{2l} + 1} \right)}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{w^{2}(r)}{\sin\left\lbrack {\frac{2\pi}{N}\left( {{2v} - \left( {{2l} + 1} \right)} \right)\left( {r + n_{0}} \right)} \right\rbrack}}}}}}}}\mspace{14mu}} & (40) \\{{{where}\mspace{14mu} v} \in \left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack} & \;\end{matrix}$

Equation 40 can be rewritten as a summation of two modified convolutionoperations of two functions h_(I,III) and h_(II) with two sets ofintermediate spectral components m_(I,III) and m_(II) that are derivedfrom the MDCT coefficients X_(I), X_(II), and XIII for three segments ofthe source signal as follows:

$\begin{matrix}\begin{matrix}{{S\left( {2v} \right)} = {{\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{{m_{I,{III}}(p)}{h_{I,{III}}\left( {{2v} - p} \right)}}}} +}} \\{{\frac{4}{N}{\sum\limits_{l = 0}^{\frac{N}{2} - 1}\;{{m_{II}\left( {{2l} + 1} \right)}{h_{II}\left( {{2v} - \left( {{2l} + 1} \right)} \right)}}}},{where}}\end{matrix} & \left( {41a} \right) \\{{m_{I,{III}}(\tau)} = \left\lbrack {{\left( {- 1} \right)^{\tau + 1}{X_{I}(\tau)}} + {X_{III}(\tau)}} \right\rbrack} & \; \\{{m_{II}(\tau)} = {X_{II}(\tau)}} & \; \\{{{h_{I,{III}}(\tau)} = {\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{\omega(r)}{\omega\left( {r + \frac{N}{2}} \right)}{\cos\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack}}}},} & \; \\{{{h_{II}(\tau)} = {\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{\omega^{2}(r)}{\sin\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack}}}},} & \; \\{{v \in \left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack},} & \; \\{{S\left( {{2v} + 1} \right)} = {S\left( {N - {2\left( {1 + v} \right)}} \right)}} & \left( {41b} \right)\end{matrix}$

The results of the modified convolution operations depend on theproperties of the functions h_(I,III) and h_(II), which are impulseresponses of hypothetical filters that are related to the combinedeffects of the IMDCT synthesis filter bank, the subsequent MDST analysisfilter bank, and the analysis and synthesis window functions Themodified convolutions need to be evaluated only for even integers.

Each of the impulse responses is symmetric. It may be seen frominspection that h_(I,III)(τ)=h_(I,III)(−τ) and h_(II)(τ)=−h_(II)(−τ).These symmetry properties may be exploited in practical digitalimplementations to reduce the amount of memory needed to store arepresentation of each impulse response. An understanding of how thesymmetry properties of the impulse responses interact with the symmetryproperties of the intermediate spectral components m_(I,III) and m_(II)may also be exploited in practical implementations to reducecomputational complexity.

The impulse responses h_(I,III)(τ) and h_(II)(τ) may be calculated fromthe summations shown above; however, it may be possible to simplifythese calculations by deriving simpler analytical expressions for theimpulse responses. Because the impulse responses depend on the windowfunction w(r), the derivation of simpler analytical expressions requiresadditional specifications for the window function. An example ofderivations of simpler analytical expressions for the impulse responsesfor two specific window functions, the rectangular and sine windowfunctions, are discussed below.

2. Rectangular Window Function

The rectangular window function is not often used in coding applicationsbecause it has relatively poor frequency selectivity properties;however, its simplicity reduces the complexity of the analysis needed toderive a specific implementation. For this derivation, the windowfunction

${w(r)} = \frac{1}{\sqrt{2}}$for r ε[0,N−1] is used. For this particular window function, the secondterm of equation 41a is equal to zero. The calculation of the MDSTcoefficients does not depend on the MDCT coefficients for the secondsegment. As a result, equation 41a may be rewritten as

$\begin{matrix}{{S\left( {2v} \right)} = {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{{m_{I,{III}}(p)}{h_{I,{III}}\left( {{2v} - p} \right)}}}}} & (42) \\{{m_{I,{III}}(\tau)} = \left\lbrack {{\left( {- 1} \right)^{\tau + 1}{X_{I}(\tau)}} + {X_{III}(\tau)}} \right\rbrack} & \; \\{{m_{II}(\tau)} = {X_{II}(\tau)}} & \; \\{{{h_{I,{III}}(\tau)} = {\frac{1}{2}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\cos\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack}}}},} & \; \\{v \in \left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack} & \;\end{matrix}$

If N is restricted to have a value that is a multiple of four, thisequation can be simplified further by using another lemma thatpostulates the following equality:

$\begin{matrix}{{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\cos\left\lbrack {\frac{2\pi}{N}(q)\left( {r + n_{0}} \right)} \right\rbrack}} = \left\{ {{\begin{matrix}{\left( {- 1} \right)^{q}\frac{\sin\frac{\pi q}{2}}{\sin\frac{\pi q}{N}}} & {q\mspace{14mu}{not}\mspace{14mu} a\mspace{14mu}{multiple}\mspace{14mu}{of}\mspace{14mu} N} \\{\left( {- 1} \right)^{\frac{q}{N}} \cdot \frac{N}{2}} & {q,{a\mspace{14mu}{multiple}\mspace{14mu}{of}\mspace{14mu} N}}\end{matrix}{where}\mspace{14mu} n_{0}} = \frac{\frac{N}{2} + 1}{2}} \right.} & (43)\end{matrix}$

This may be proven as follows:

$\begin{matrix}\begin{matrix}{I = {{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\cos\left\lbrack {\frac{2\pi}{N}(q)\left( {r + n_{0}} \right)} \right\rbrack}} = {\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\cos\left\lbrack {{\frac{2\pi}{N}(q)\left( {r + n_{0}} \right)} + \frac{\pi}{2}} \right\rbrack}}}} \\{= {\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\sin\left\lbrack {{\frac{2\pi}{N}(q)\left( {r + n_{0}} \right)} + {\frac{2\pi}{N}(q)\left( \frac{N}{4q} \right)}} \right\rbrack}}} \\{= {\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\sin\left\lbrack {\frac{2\pi}{N}(q)\left( {r + n_{0} + \frac{N}{4q}} \right)} \right\rbrack}}}\end{matrix} & (44)\end{matrix}$By using the lemma shown in equation 35 with

${a = {n_{0} + \frac{N}{4q}}},$expression 44 can be rewritten as

$\begin{matrix}{I = {{\sin\left( {\frac{2{{\pi q}\left( {n_{0} + \frac{N}{4q}} \right)}}{N} = {\frac{\pi q}{2} - \frac{piq}{N}}} \right)}\frac{\sin\frac{\pi q}{2}}{\sin\frac{\pi q}{N}}}} & (45)\end{matrix}$which can be simplified to obtain the following expression:

$\begin{matrix}{I = {\left( {- 1} \right)^{q}\frac{\sin\frac{\pi q}{2}}{\sin\frac{\pi q}{N}}}} & (46)\end{matrix}$

If q is an integer multiple of N such that q=mN, then the numerator anddenominator of the quotient in expression 46 are both equal to zero,causing the value of the quotient to be indeterminate. L'Hospital's rulemay be used to simplify the expression further. Differentiating thenumerator and denominator with respect to q and substituting q=mN yieldsthe expression

$\frac{N \cdot {\cos\left( \frac{\pi mN}{2} \right)}}{2 \cdot {\cos({\pi m})}}$Because N is an integer multiple of four, the numerator is always equalto N and the denominator is equal to 2·(−1)^(m)=2·(−1)^(q/N). Thiscompletes the proof of the lemma expressed by equation 43.

This equality may be used to obtain expressions for the impulse responseh_(I,III). Different cases are considered to evaluate the responseh_(I,III)(τ). If τ is an integer multiple of N such that τ=mN thenh_(I,III)(τ)=(−1)^(m)·N/4. The response equals zero for even values of τother than an integer multiple of N because the numerator of thequotient in equation 46 is equal to zero. The value of the impulseresponse h_(I,III) for odd values of τ can be seen from inspection. Theimpulse response may be expressed as follows:

$\begin{matrix}\begin{matrix}{{h_{I,{III}}(\tau)} = {{\left( {- 1} \right)^{m}\frac{N}{4}\mspace{14mu}{for}\mspace{14mu}\tau} = {mN}}} \\{{{h_{I,{III}}(\tau)} = {0\mspace{14mu}{for}\mspace{14mu}\tau\mspace{14mu}{even}}},{\tau \neq 0}}\end{matrix} & (47) \\{{h_{I,{III}}(\tau)} = {\frac{1}{2}\frac{\left( {- 1} \right)^{\frac{\tau + 1}{2}}}{\sin\frac{\pi\tau}{N}}}} & (48)\end{matrix}$The impulse response h_(I,III) for a rectangular window function andN=128 is illustrated in FIG. 6.

By substituting these expressions into equation 42, equations 41a and41b can be rewritten as:

$\begin{matrix}{{S\left( {2v} \right)} = {\frac{2}{N}{\sum\limits_{p = 0}^{N - 1}\;{{m_{I,{III}}(p)}{h_{I,{III}}\left( {{2v} - p} \right)}}}}} & \left( {49a} \right) \\{{m_{I,{III}}(\tau)} = \left\lbrack {{\left( {- 1} \right)^{\tau + 1}{X_{I}(\tau)}} + {X_{III}(\tau)}} \right\rbrack} & \; \\{{h_{I,{III}}(\tau)} = \left\{ \begin{matrix}{{\left( {- 1} \right)^{m}\frac{N}{4}},{\tau = {mN}}} \\{0,{\tau \neq {{mN}\mspace{14mu}{and}\mspace{14mu}\tau\mspace{14mu}{even}}}} \\{{\frac{1}{2} \cdot \frac{\left( {- 1} \right)^{\frac{\tau + 1}{2}}}{\sin^{\frac{3\tau}{N}}}},{\tau\mspace{14mu}{odd}}}\end{matrix} \right.} & \; \\{{S\left( {{2v} + 1} \right)} = {{{S\left( {N - {2\left( {1 + v} \right)}} \right)}v} \in \left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack}} & \left( {49b} \right)\end{matrix}$

Using equations 49a and 49b, MDST coefficients for segment II can becalculated from the MDCT coefficients of segments I and III assuming theuse of a rectangular window function. The computational complexity ofthis equation can be reduced by exploiting the fact that the impulseresponse h_(I,III)(τ) is equal to zero for many odd values of τ.

3. Sine Window Function

The sine window function has better frequency selectivity propertiesthan the rectangular window function and is used in some practicalcoding systems. The following derivation uses a sine window functiondefined by the expressionw(r)=sin(π/N(r+½))  (50)

A simplified expression for the impulse response h_(I,III) may bederived by using a lemma that postulates the following:

$\begin{matrix}\begin{matrix}{{{I(\tau)} = {\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\omega(r){\omega\left( {r + \frac{N}{2}} \right)}{\cos\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack}}}},} \\{= \left\{ \begin{matrix}{0,{\tau\mspace{14mu}{odd}},{\tau \neq {{mN} + 1}},{\tau \neq {{mN} - 1}}} \\{{{- \frac{N}{S}}{\left( {- 1} \right)^{m} \cdot \tau}} = {{mN} + 1}} \\{{{- \frac{N}{S}}{\left( {- 1} \right)^{m} \cdot \tau}} = {{mN} + 1}} \\{{\frac{\left( {- 1} \right)^{\frac{3\tau}{2}}}{4}\left\lbrack {\frac{1}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)} + \frac{1}{\sin\frac{\pi}{N}\left( {{- \tau} + 1} \right)}} \right\rbrack},{\tau\mspace{14mu}{even}}}\end{matrix} \right.}\end{matrix} & (51) \\{{{where}\mspace{14mu}{\omega(r)}} = {\sin\left( {\frac{\pi}{N}\left( {r + \frac{1}{2}} \right)} \right)}} & \;\end{matrix}$

This lemma may be proven by first simplifying the expression forw(r)w(r+N/2) as follows:

$\begin{matrix}{{{\sin\left( {\frac{\pi}{N}\left( {r + \frac{1}{2}} \right)} \right)}{\sin\left( {\frac{\pi}{N}\left( {r + \frac{N}{2} + \frac{1}{2}} \right)} \right)}} = {{{\sin\left( {\frac{\pi}{N}\left( {r + \frac{1}{2}} \right)} \right)}{\sin\left( {{\frac{\pi}{N}\left( {r + \frac{1}{2}} \right)} + \frac{\pi}{2}} \right)}} = {{{\sin\left( {\frac{\pi}{N}\left( {r + \frac{1}{2}} \right)} \right)}{\cos\left( {\frac{\pi}{N}\left( {r + \frac{1}{2}} \right)} \right)}} = {\frac{1}{2}{\sin\left( {\frac{2\pi}{N}\left( {r + \frac{1}{2}} \right)} \right.}}}}} & (52)\end{matrix}$Substituting this simplified expression into equation 51 obtains thefollowing:

$\begin{matrix}{{I(\tau)} = {\frac{1}{2}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{\sin\left\lbrack {\frac{2\pi}{N}\left( {r + \frac{1}{2}} \right)} \right\rbrack}{\cos\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack}}}}} & (53)\end{matrix}$

Using the following trigonometric identitysin u cos v=½[sin(u+v)+sin(u−v)]  (54)equation 53 can be rewritten as follows:

$\begin{matrix}{{I(\tau)} = {{\frac{1}{4}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\sin\left\lbrack {{\frac{2\pi}{N}\left( {r + \frac{1}{2}} \right)} + {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)}} \right\rbrack}}} + {\frac{1}{4}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\sin\left\lbrack {{\frac{2\pi}{N}\left( {r + \frac{1}{2}} \right)} + {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)}} \right\rbrack}}}}} & (55) \\{{I(\tau)} = {{\frac{1}{4}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\sin\left\lbrack {{\frac{2\pi}{N}\left( {r + \frac{1}{2}} \right)} + {\tau r} + {\tau n}_{0}} \right\rbrack}}} + {\frac{1}{4}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\sin\left\lbrack {\frac{2\pi}{N}\left( {{\left( {{- \tau} + 1} \right)r} - \left( {{\tau n}_{0} - \frac{1}{2}} \right)} \right)} \right\rbrack}}}}} & \; \\{{I(\tau)} = {{\frac{1}{4}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\sin\left\lbrack {\frac{2\pi}{N}\left( {{\left( {\tau + 1} \right)r} + \left( {{\tau n}_{0} + \frac{1}{2}} \right)} \right)} \right\rbrack}}} + {\frac{1}{4}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\sin\left\lbrack {\frac{2\pi}{N}\left( {{\left( {{- \tau} + 1} \right)r} - \left( {{\tau n}_{0} - \frac{1}{2}} \right)} \right)} \right\rbrack}}}}} & \; \\{{I(\tau)} = {{\frac{1}{4}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\sin\left\lbrack {\frac{2\pi}{N}\left( {\tau + 1} \right)\left( {r + \frac{{\tau n}_{0} + \frac{1}{2}}{\tau + 1}} \right)} \right\rbrack}}} + {\frac{1}{4}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\sin\left\lbrack {\frac{2\pi}{N}\left( {{- \tau} + 1} \right)\left( {r - \frac{{\tau n}_{0} - \frac{1}{2}}{{- \tau} + 1}} \right)} \right\rbrack}}}}} & (56)\end{matrix}$

Equation 55 can be simplified by substitution in both terms of I(τ)according to equation 35, setting q=(τ+1) and

$a = \frac{{r\; n_{0}} + \frac{1}{2}}{\left( {\tau + 1} \right)}$in the first term, and setting q=(−τ+1) and

$a = \frac{{r\; n_{0}} - \frac{1}{2}}{\left( {{- \tau} + 1} \right)}$in the second term. This yields the following:

$\begin{matrix}{{I(\tau)} = {{\frac{1}{4}{\sin\left( {{\frac{2\pi}{N}\left( {{\tau n}_{0} + \frac{1}{2}} \right)} + {\frac{\pi}{2}\left( {\tau + 1} \right)} - {\frac{\pi}{N}\left( {\tau + 1} \right)}} \right)}\frac{\sin\frac{\pi}{2}\left( {\tau + 1} \right)}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)}} + {\frac{1}{4}{\sin\left( {{\frac{2\pi}{N}\left( {{- {\tau n}_{0}} + \frac{1}{2}} \right)} + {\frac{\pi}{2}\left( {{- \tau} + 1} \right)} - {\frac{\pi}{N}\left( {{- \tau} + 1} \right)}} \right)}\frac{\sin\frac{\pi}{2}\left( {{- \tau} + 1} \right)}{\sin\frac{\pi}{N}\left( {{- \tau} + 1} \right)}}}} & (57) \\{{I(\tau)} = {{\frac{1}{4}{\sin\left( {{\frac{\pi}{N}(\tau)\left( {\frac{N}{2} + 1} \right)} + {\frac{\pi}{2}\left( {\tau + 1} \right)} - {\frac{\pi}{N}(\tau)}} \right)}\frac{\sin\frac{\pi}{2}\left( {\tau + 1} \right)}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)}} + {\frac{1}{4}{\sin\left( {{\frac{\pi}{N}\left( {- \tau} \right)\left( {\frac{N}{2} + 1} \right)} + {\frac{\pi}{2}\left( {{- \tau} + 1} \right)} - {\frac{\pi}{N}\left( {- \tau} \right)}} \right)}\frac{\sin\frac{\pi}{2}\left( {{- \tau} + 1} \right)}{\sin\frac{\pi}{N}\left( {{- \tau} + 1} \right)}}}} & \; \\{{I(\tau)} = {{\frac{1}{4}\left( {{\frac{\pi}{2}(\tau)} + {\frac{\pi}{2}\left( {\tau + 1} \right)}} \right)\frac{\sin\frac{\pi}{2}\left( {\tau + 1} \right)}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)}} + {\frac{1}{4}\left( {{\frac{\pi}{2}\left( {- \tau} \right)} + {\frac{\pi}{2}\left( {{- \tau} + 1} \right)}} \right)\frac{\sin\frac{\pi}{2}\left( {{- \tau} + 1} \right)}{\sin\frac{\pi}{N}\left( {{- \tau} + 1} \right)}}}} & \; \\{{I(\tau)} = {{\frac{1}{4}{\sin\left( {{\pi(\tau)} + \frac{\pi}{2}} \right)}\frac{\sin\frac{\pi}{2}\left( {\tau + 1} \right)}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)}} + {\frac{1}{4}{\sin\left( {{\pi\left( {- \tau} \right)} + \frac{\pi}{2}} \right)}\frac{\sin\frac{\pi}{2}\left( {{- \tau} + 1} \right)}{\sin\frac{\pi}{N}\left( {{- \tau} + 1} \right)}}}} & \; \\{\mspace{79mu}{{I(\tau)} = {{\frac{1}{4}{{\cos({\pi\tau})} \cdot \frac{\cos\frac{\pi}{2}\tau}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)}}} + {\frac{1}{4}{{\cos\left( {- {\pi\tau}} \right)} \cdot \frac{\cos\frac{- \pi}{2}\tau}{\sin\frac{- \pi}{N}\left( {{- \tau} + 1} \right)}}}}}} & (58) \\{\mspace{79mu}{{I(\tau)} = {{\frac{\left( {- 1} \right)^{T}}{4} \cdot \frac{\cos\frac{\pi}{2}\tau}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)}} + {\frac{\left( {- 1} \right)^{T}}{4} \cdot \frac{\cos\frac{\pi}{2}\left( {- \tau} \right)}{\sin\frac{- \pi}{N}\left( {{- \tau} + 1} \right)}}}}} & \; \\{\mspace{76mu}{{I(\tau)} = {\frac{\left( {- 1} \right)^{T}}{4}\cos\frac{\pi}{2}{\tau \cdot \left\lbrack {\frac{1}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)} + \frac{1}{\sin\frac{\pi}{N}\left( {{- \tau} + 1} \right)}} \right\rbrack}}}} & \; \\{\mspace{76mu}{{I(\tau)} = \left\{ \begin{matrix}{{\frac{\left( {- 1} \right)^{\frac{3_{\tau}}{2}}}{4}\left\lbrack {\frac{1}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)} + \frac{1}{\sin\frac{\pi}{N}\left( {{- \tau} + 1} \right)}} \right\rbrack},} & {\tau\mspace{14mu}{even}} \\{0,} & {\tau\mspace{14mu}{odd}}\end{matrix} \right.}} & \;\end{matrix}$

Equation 58 is valid unless the denominator for either quotient is equalto zero. These special cases can be analyzed by inspecting equation 57to identify the conditions under which either denominator is zero. Itcan be seen from equation 57 that singularities occur for τ=mN+1 andτ=mN−1, where m is an integer. The following assumes N is an integermultiple of four.

For τ=mN+1 equation 57 can be rewritten as:

$\begin{matrix}\begin{matrix}{{I\left( {{mN} + 1} \right)} = {{\frac{1}{4}{{\sin\left( {{\pi\left( {{mN} + 1} \right)} + \frac{\pi}{2}} \right)} \cdot \frac{\sin\frac{\pi}{2}\left( {{mN} + 2} \right)}{\sin\frac{\pi}{N}\left( {{mN} + 1} \right)}}} +}} \\{\frac{1}{4}{{\sin\left( {{- {\pi\left( {{mN} + 1} \right)}} + \frac{\pi}{2}} \right)} \cdot \frac{\sin\frac{\pi}{2}\left( {{- \left( {{mN} + 1} \right)} + 1} \right)}{\sin\frac{\pi}{2}\left( {{- \left( {{mN} + 1} \right)} + 1} \right)}}} \\{= {0 + {\frac{1}{4}{\sin\left( {{- {\pi{mN}}} - \frac{\pi}{2}} \right)}\frac{\sin\frac{{- {mN}}\;\pi}{2}}{\sin\frac{{- {mN}}\;\pi}{N}}}}} \\{= {{- \frac{1}{4}}\frac{\sin\frac{{- {mN}}\;\pi}{2}}{\sin\frac{{- {mN}}\;\pi}{N}}}}\end{matrix} & (59)\end{matrix}$The value of the quotient is indeterminate because the numerator anddenominator are both equal to zero. L'Hospital's rule can be used todetermine its value. Differentiating numerator and denominator withrespect to m yields the following:

$\begin{matrix}\begin{matrix}{{I\left( {{mN} + 1} \right)} = {{- \frac{1}{4}}\frac{{- \frac{N\;\pi}{2}}\cos\frac{{- {mN}}\;\pi}{2}}{{- {\pi cos}} - {mn}}}} \\{= {{- \frac{N}{8}}\left( {- 1} \right)^{m}}}\end{matrix} & (60)\end{matrix}$

For τ=mN−1 equation 57 can be rewritten as:

$\begin{matrix}{{I\left( {{mN} - 1} \right)} = {{\frac{1}{4}{{\sin\left( {{\pi\left( {{mN} - 1} \right)} + \frac{\pi}{2}} \right)} \cdot \frac{\sin\frac{\pi}{2}\left( {{mN} + 1 - 1} \right)}{\sin\frac{\pi}{2}\left( {{mN} + 1 - 1} \right)}}} + {\frac{1}{4}{{\sin\left( {{- {\pi\left( {{mN} - 1} \right)}} + \frac{\pi}{2}} \right)} \cdot \frac{\sin\frac{\pi}{2}\left( {- \left( {{mN} - 1 + 1} \right)} \right.}{\sin\frac{\pi}{2}\left( {- \left( {{mN} - 1 + 1} \right)} \right.}}}}} & (61) \\{\mspace{76mu}{{I\left( {{mN} - 1} \right)} = {{\frac{1}{4}{\sin\left( {{\pi{mN}} - \frac{\pi}{2}} \right)}\frac{\sin\frac{{\pi mN}\;}{2}}{\sin\frac{{\pi mN}\;}{2}}} + 0}}} & \;\end{matrix}$The value of the quotient in this equation is indeterminate because thenumerator and denominator are both equal to zero. L'Hospital's rule canbe used to determine its value. Differentiating numerator anddenominator with respect to m yields the following:

$\begin{matrix}\begin{matrix}{{I\left( {{mN} - 1} \right)} = {{- \frac{1}{4}}{\frac{\frac{\pi N}{2}}{\frac{\pi}{N}} \cdot \frac{\cos\frac{{\pi mN}\;}{2}}{\cos\;{\pi m}}}}} \\{= {{- \frac{N\;}{8}}\left( {- 1} \right)^{m}}}\end{matrix} & (62)\end{matrix}$

The lemma expressed by equation 51 is proven by combining equations 58,60 and 62.

A simplified expression for the impulse response h_(II) may be derivedby using a lemma that postulates the following:

$\begin{matrix}\begin{matrix}{{{I(\tau)} = {\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{{\omega(r)}{\omega(r)}{\sin\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack}}}},} \\{= \left\{ \begin{matrix}{0,{\tau\mspace{14mu}{odd}},{\tau \neq {{mN} + 1}},{\tau \neq {{mN} - 1 -}}} \\{{\frac{N}{S}\left( {- 1} \right)^{m}},{\tau = {{mN} + 1 -}}} \\{{\frac{N}{S}\left( {- 1} \right)^{m + 1}},{\tau = {{mN} - 1}}} \\\begin{matrix}{{\frac{\left( {- 1} \right)^{\frac{\delta\;\tau}{2}}}{4}\left\lbrack {\frac{- 1}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)} + \frac{1}{\sin\frac{\pi}{N}\left( {{- \tau} + 1} \right)}} \right\rbrack},} & {\tau\mspace{14mu}{even}\mspace{14mu}{where}}\end{matrix}\end{matrix} \right.}\end{matrix} & (63) \\{{\omega(r)} = {\sin\left( {\frac{\pi}{N}\left( {r + \frac{1}{2}} \right)} \right)}} & \;\end{matrix}$

The proof of this lemma is similar to the previous proof. This proofbegins by simplifying the expression for w(r)w(r). Recall that sin²α=½−½ cos (2a), so that:

$\begin{matrix}{{\sin^{2}\left( {\frac{\pi}{N}\left( {r + \frac{1}{2}} \right)} \right)} = {\frac{1}{2} - {\frac{1}{2}{\cos\left( {\frac{2\pi}{N}\left( {r + \frac{1}{2}} \right)} \right)}}}} & (64)\end{matrix}$Using this expression, equation 63 can be rewritten as:

$\begin{matrix}\begin{matrix}{{I(\tau)} = {\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\left\lbrack {\frac{1}{2} - {\frac{1}{2}{\cos\left( {\frac{2\pi}{N}\left( {r + \frac{1}{2}} \right)} \right)}}} \right\rbrack{\sin\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack}}}} \\{= {{\frac{1}{2}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\sin\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack}}} -}} \\{\frac{1}{2}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{\cos\left\lbrack {\frac{2\pi}{N}\left( {r + \frac{1}{2}} \right)} \right\rbrack}{\sin\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack}}}}\end{matrix} & (65)\end{matrix}$

From equation 37 and the associated lemma, it may be seen the first termin equation 65 is equal to zero. The second term may be simplified usingthe trigonometric identity cos u·sin v=½[ sin (u+v)−sin (u−v)], whichobtains the following:

$\begin{matrix}{{I(\tau)} = {{{- \frac{1}{4}}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}\;{\sin\left\lbrack {{\frac{2\pi}{N}\left( {r + \frac{1}{2}} \right)} + {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)}} \right\rbrack}}} + {\frac{1}{4}{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{\sin\left\lbrack {{\frac{2\pi}{N}\left( {r + \frac{1}{2}} \right)} - {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)}} \right\rbrack}}}}} & (66)\end{matrix}$

Referring to equation 66, its first term is equal to the negative of thefirst term in equation 55 and its second term is equal to the secondterm of equation 55. The proof of the lemma expressed in equation 63 maybe proven in a manner similar to that used to prove the lemma expressedin equation 51. The principal difference in the proof is the singularityanalyses of equation 59 and equation 61. For this proof, I(mN−1) ismultiplied by an additional factor of −1; therefore,

${I\left( {{mN} - 1} \right)} = {\frac{N\;}{8}{\left( {- 1} \right)^{m + 1}.}}$Allowing for this difference along with the minus sign preceding thefirst term of equation 55, the lemma expressed in equation 63 is proven.

An exact expression for impulse response h_(ll)(τ) is given by thislemma; however, it needs to be evaluated only for odd values of τbecause the modified convolution of h_(II) in equation 41a is evaluatedonly for τ=(2v−(2l+1)). According to equation 63, h_(II)(τ)=0 for oddvalues of τ except for τ=mN+1 and τ=mN−1. Because h_(II)(τ) is non-zerofor only two values of τ, this impulse response can be expressed as:

$\begin{matrix}{{h_{II}(\tau)} = \left\{ \begin{matrix}{{{- \frac{N}{S}}\left( {- 1} \right)^{m}},} & {\tau = {{mN} + 1}} \\{{{- \frac{N}{S}}\left( {- 1} \right)^{m + 1}},} & {\tau = {{mN} - 1}} \\{0,} & {otherwise}\end{matrix} \right.} & (67)\end{matrix}$The impulse responses h_(I,III)(τ) and h_(II)(τ) for the sine windowfunction and N=128 are illustrated in FIGS. 7 and 8, respectively.

Using the analytical expressions for the impulse responses h_(I,III) andh_(II) provided by equations 51 and 67, equations 41a and 41b can berewritten as:

$\begin{matrix}\begin{matrix}{{S\left( {2v} \right)} = {{\frac{2}{N}{\sum\limits_{p = 0}^{\frac{N}{2} - 1}{{m_{I,{III}}(p)}{h_{I,{III}}\left( {{2v} - p} \right)}}}} +}} \\{{\frac{4}{N}{\sum\limits_{l = 0}^{\frac{N}{2} - 1}{{m_{II}\left( {{2l} + 1} \right)}{h_{II}\left( {{2v} - \left( {{2l} + 1} \right)} \right)}}}},{where}}\end{matrix} & \left( {68a} \right) \\{{m_{I,{III}}(\tau)} = \left\lbrack {{\left( {- 1} \right)^{\tau + 1}{X_{I}(\tau)}} + {X_{III}(\tau)}} \right\rbrack} & \; \\{{m_{II}(\tau)} = {X_{II}(\tau)}} & \; \\{{h_{I,{III}}(\tau)} = \left\{ \begin{matrix}{0,{\tau\mspace{14mu}{odd}},{\tau \neq {{mN} + 1}},{\tau \neq {{mN} - 1 -}}} \\{{\frac{N}{S}\left( {- 1} \right)^{m}},{\tau = {{mN} + 1 -}}} \\{{\frac{N}{S}\left( {- 1} \right)^{m}},{\tau = {{mN} - 1}}} \\\begin{matrix}{{\frac{\left( {- 1} \right)^{\frac{\delta\;\tau}{2}}}{4}\left\lbrack {\frac{1}{\sin\frac{\pi}{N}\left( {\tau + 1} \right)} + \frac{1}{\sin\frac{\pi}{N}\left( {{- \tau} + 1} \right)}} \right\rbrack},} & {\tau\mspace{14mu}{even}}\end{matrix}\end{matrix} \right.} & \; \\{{h_{II}(\tau)} = \left\{ \begin{matrix}{{{- \frac{N}{S}}\left( {- 1} \right)^{m}},} & {\tau = {{mN} + 1}} \\{{{- \frac{N}{S}}\left( {- 1} \right)^{m + 1}},} & {\tau = {{mN} - 1}} \\{0,} & {otherwise}\end{matrix} \right.} & \; \\{{S\left( {{2v} + 1} \right)} = {S\left( {N - {2\left( {1 + v} \right)}} \right)}} & \left( {68b} \right)\end{matrix}$

Using equations 68a and 68b, MDST coefficients for segment II can becalculated from the MDCT coefficients of segments I, II and III assumingthe use of a sine window function. The computational complexity of thisequation can be reduced further by exploiting the fact that the impulseresponse h_(I,III)(τ) is equal to zero for many odd values of τ.

C. Spectral Component Estimation

Equations 41a and 41b express a calculation of exact MDST coefficientsfrom MDCT coefficients for an arbitrary window function. Equations 49a,49b, 68a and 68b express calculations of exact MDST coefficients fromMDCT coefficients using a rectangular window function and a sine windowfunction, respectively. These calculations include operations that aresimilar to the convolution of impulse responses. The computationalcomplexity of calculating the convolution-like operations can be reducedby excluding from the calculations those values of the impulse responsesthat are known to be zero.

The computational complexity can be reduced further by excluding fromthe calculations those portions of the full responses that are of lessersignificance; however, this resulting calculation provides only anestimate of the MDST coefficients because an exact calculation is nolonger possible. By controlling the amounts of the impulse responsesthat are excluded from the calculations, an appropriate balance betweencomputational complexity and estimation accuracy can be achieved.

The impulse responses themselves are dependent on the shape of thewindow function that is assumed. As a result, the choice of windowfunction affects the portions of the impulse responses that can beexcluded from calculation without reducing coefficient estimationaccuracy below some desired level.

An inspection of equation 49a for rectangular window functions shows theimpulse response h_(I,III) is symmetric about τ=0 and decays moderatelyrapidly. An example of this impulse response for N=128 is shown in FIG.6. The impulse response h_(II) is equal to zero for all values of τ.

An inspection of equation 68a for the sine window function shows theimpulse response h_(I,III) is symmetric about τ=0 and decays morerapidly than the corresponding response for the rectangular windowfunction. For the sine window function, the impulse response h_(II) isnon-zero for only two values of τ. An example of the impulse responsesh_(I,III) and h_(II) for a sine window function and N=128 are shown inFIGS. 7 and 8, respectively.

Based on these observations, a modified form of equations 41a and 41bthat provides an estimate of MDST coefficients for any analysis orsynthesis window function may be expressed in terms of two filterstructures as follows:

$\begin{matrix}{{S\left( {2v} \right)} = {{{filter\_ structure}\_ 1\left( {2v} \right)} + {{filter\_ structure}\_ 2\left( {2v} \right)}}} & (69) \\{{{filter\_ structure}\_ 1\left( {2v} \right)} = {\frac{2}{N}{\sum\limits_{p = 0}^{\frac{N}{2} - 1}{{m_{I,{III}}(p)}{h_{I,{III}}\left( {{2v} - p} \right)}}}}} & (70) \\{{m_{I,{III}}(\tau)} = \left\lbrack {{\left( {- 1} \right)^{\tau + 1}{X_{I}(\tau)}} + {X_{III}(\tau)}} \right\rbrack} & (71) \\{{h_{I,{III}}(\tau)} = \left\{ \begin{matrix}{{0\mspace{14mu}{if}\mspace{14mu}\tau}\; \in \left\lbrack {\tau_{{trunc}\; 1},{N - \tau_{{trunc}\; 1}}} \right\rbrack} \\{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{\omega(\tau)}{{\omega\left( {r + \frac{N}{2}} \right)}.}}} \\{{\cos\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack},{o\text{.}\omega\text{.}}}\end{matrix} \right.} & (72) \\{{{filter\_ structure}\_ 2\left( {2v} \right)} = {\frac{4}{N}{\sum\limits_{l = 0}^{\frac{N}{2} - 1}{{m_{II}\left( {{2l} + 1} \right)}{h_{II}\left( {{2v} - \left( {{2l} + 1} \right)} \right)}}}}} & (73) \\{{m_{II}(\tau)} = {X_{II}(\tau)}} & (74) \\{{h_{II}(\tau)} = \left\{ \begin{matrix}{{0\mspace{14mu}{if}\mspace{14mu}\tau}\; \in \left\lbrack {\tau_{{trunc}\; 2},{\frac{N}{2} - 1 - \tau_{{trunc}\; 2}}} \right\rbrack} \\{\sum\limits_{r = 0}^{\frac{N}{2} - 1}{{\omega^{2}(r)}{\sin\left\lbrack {\frac{2\pi}{N}(\tau)\left( {r + n_{0}} \right)} \right\rbrack}}}\end{matrix} \right.} & (75) \\{{{S\left( {{2v} + 1} \right)} = {S\left( {N - {2\left( {1 + v} \right)}} \right)}}{where}} & (76) \\{{{v \in \left\lbrack {0,{\frac{N}{2} - 1}} \right\rbrack},{n_{0} = {\frac{\frac{N}{2} + 1}{2}\mspace{14mu}{and}}}}\text{}{{ntaps}_{tot},\tau_{{trunc}\; 1},{\tau_{{trunc}\; 2}\mspace{14mu}{are}\mspace{14mu}{chosen}\mspace{14mu}{to}\mspace{14mu}{satisfy}}}} & (77) \\{{\tau_{{tru}\;{nc}\; 1} \in \left\lbrack {1,\frac{N}{2}} \right\rbrack},{\tau_{{trunc}\; 2} \in \left\lbrack {1,{\frac{N}{4} - 1}} \right\rbrack},\text{}{{ntaps}_{tot} = {{2\tau_{{trunc}\; 1}} - 1 + {2\tau_{{trunc}\; 2}}}}} & (78)\end{matrix}$

An example of a device 30 that estimates MDST coefficients according toequation 69 is illustrated by a schematic block diagram in FIG. 3. Inthis implementation, the intermediate component generator 32 receivesMDCT coefficients from the path 1 and derives first intermediatecomponents m_(I,III) from the MDCT coefficients X_(I) and X_(II) ofsegments I and III, respectively, by performing the calculations shownin equation 71, and derives first intermediate components m_(II) fromthe MDCT coefficients X_(II) of segment II by performing thecalculations shown in equation 74. The intermediate component generator34 derives second intermediate components by forming a combination offirst intermediate components m_(I,III) according to a portion of theimpulse response h_(I,III) received from the impulse responses 33 byperforming the calculations shown in equation 70, and derives secondintermediate components by forming a combination of first intermediatecomponents m_(II) according to a portion of the impulse response h_(II)received from the impulse responses 33 by performing the calculationsshown in equation 73. Any portion of the two impulse responses may beused as expressed by the values τ_(trunc1) and τ_(trunc2) including theentire responses. The use of longer impulse responses increasescomputational complexity and generally increases the accuracy of MDSTcoefficient estimation. The spectral component generator 35 obtains MDSTcoefficients from the second intermediate components by performing thecalculations shown in equations 69 and 76.

The magnitude and phase estimator 36 calculates measures of magnitudeand phase from the calculated MDST coefficients and the MDCTcoefficients received from the path 31 and passes these measures alongthe paths 38 and 39. The MDST coefficients may also be passed along thepath 37. Measures of spectral magnitude and phase may be obtained byperforming the calculations shown above in equations 10 and 11, forexample. Other examples of measures that may be obtained includespectral flux, which may be obtained from the first derivative ofspectral magnitude, and instantaneous frequency, which may be obtainedfrom the first derivative of spectral phase.

Referring to the impulse responses shown in FIGS. 6–8, for example, itmay be seen that the coefficient values obtained by the convolution-typeoperations of the two filter structures are dominated by the portions ofthe responses that are near τ=0. A balance between computationalcomplexity and estimation accuracy may be achieved for a particularimplementation by choosing the total number of filter taps ntaps_(tot)that are used to implement the two filter structures. The total numberof taps ntaps_(tot) may be distributed between the first and secondfilter structures as desired according to the values of τ_(trunc1) andτ_(trunc2), respectively, to adapt MDST coefficient estimation to theneeds of specific applications. The distribution of taps between the twofilter structures can affect estimation accuracy but it does not affectcomputational complexity.

The number and choice of taps for each filter structure can be selectedusing any criteria that may be desired. For example, an inspection oftwo impulse responses h_(I,III) and h_(II) will reveal the portions ofthe responses that are more significant. Taps may be chosen for only themore significant portions. In addition, computational complexity may bereduced by obtaining only selected MDST coefficients such as thecoefficients in one or more frequency ranges.

An adaptive implementation of the present invention may use largerportions of the impulse responses to estimate the MDST coefficients forspectral components that are judged to be perceptually more significantby a perceptual model. For example, a measure of perceptual significancefor a spectral component could be derived from the amount by which thespectral component exceeds a perceptual masking threshold that iscalculated by a perceptual model. Shorter portions of the impulseresponses may be used to estimate MDST coefficients for perceptuallyless significant spectral components. Calculations needed to estimateMDST coefficients for the least significant spectral components can beavoided.

A non-adaptive implementation may obtain estimates of MDST coefficientsin various frequency subbands of a signal using portions of the impulseresponses whose lengths vary according to the perceptual significance ofthe subbands as determined previously by an analysis of exemplarysignals. In many audio coding applications, spectral content in lowerfrequency subbands generally has greater perceptual significance thanspectral content in higher frequency subbands. In these applications,for example, a non-adaptive implementation could estimate MDSTcoefficients in subbands using portions of the impulse responses whoselength varies inversely with the frequency of the subbands.

D. Additional Considerations

The preceding disclosure sets forth examples that describes only a fewimplementations of the present invention. Principles of the presentinvention may be applied and implemented in a wide variety of ways.Additional considerations are discussed below.

1. Other Transforms

The exemplary implementations described above are derived from the MDCTthat is expressed in terms of the ODFT as applied to fixed-lengthsegments of a source signal that overlap one another by half the segmentlength. A variation of the examples discussed above as well as avariation of the alternatives discussed below may be obtained byderiving implementations from the MDST that is expressed in terms of theODFT.

Additional implementations of the present invention may be derived fromexpressions of other transforms including the DFT, the FFT and ageneralized expression of the MDCT filter bank discussed in the Princenpaper cited above. This generalized expression is described in U.S. Pat.No. 5,727,119 issued Mar. 10, 1998.

Implementations of the present invention also may be derived fromexpressions of transforms that are applied to varying-length signalsegments and transforms that are applied to segments having no overlapor amounts of overlap other than half the segment length.

2. Adaptive Estimation

Some empirical results suggest that an implementation of the presentinvention with a specified level of computational complexity is oftenable to derive measures of spectral component magnitude that is moreaccurate for spectral components representing a band of spectral energythan it is for spectral components representing a single sinusoid or afew sinusoids that are isolated from one another in frequency. Theprocess that estimates spectral component magnitude may be adapted in atleast two ways to improve estimation accuracy for signals that haveisolated spectral components.

One way to adapt the process is by adaptively increasing the length ofthe impulse responses for two filter structures shown in equation 69 sothat more accurate computations can be performed for a restricted set ofMDST coefficients that are related to the one or more isolated spectralcomponents.

Another way to adapt this process is by adaptively performing analternate method for deriving spectral component magnitudes for isolatedspectral components. The alternate method derives an additional set ofspectral components from the MDCT coefficients and the additional set ofspectral components are used to obtain measures of magnitude and/orphase. This adaptation may be done by selecting the more appropriatemethod for segments of the source signal, and it may be done by usingthe more appropriate method for portions of the spectrum for aparticular segment. A method that is described in the Merdjani papercited above is one possible alternate method. If it is used, this methodpreferably is extended to provide magnitude estimates for more than asingle sinusoid. This may be done by dynamically arranging MDCTcoefficients into bands of frequencies in which each band has a singledominant spectral component and applying the Merdjani method to eachband of coefficients.

The presence of a source signal that has one dominant spectral componentor a few isolated dominant spectral components may be detected using avariety of techniques. One technique detects local maxima in MDCTcoefficients having magnitudes that exceed the magnitudes of adjacentand nearby coefficients by some threshold amount and either counting thenumber of local maxima or determining the spectral distance betweenlocal maxima. Another technique determines the spectral shape of thesource signal by calculating an approximate Spectral Flatness Measure(SFM) of the source signal. The SFM is described in N. Jayant et al.,“Digital Coding of Waveforms,” Prentice-Hall, 1984, p. 57, and isdefined as the ratio of the geometric mean and the arithmetic mean ofsamples of the power spectral density of a signal.

3. Implementation

The present invention may be used advantageously in a wide variety ofapplications. Schematic block diagrams of a transmitter and a receiverincorporating various aspects of the present invention are shown inFIGS. 4 and 5, respectively.

The transmitter shown in FIG. 4 is similar to the transmitter shown inFIG. 1 and includes the estimator 30, which incorporates various aspectsof the present invention to provide measures of magnitude and phasealong the paths 38 and 39, respectively. The encoder 6 uses thesemeasures to generate encoded information representing the spectralcomponents received from the analysis filter bank 3. Examples ofprocesses that may be used in the encoder 6, which may depend on themeasures of magnitude or phase, include perceptual models used todetermine adaptive quantization levels, coupling, and spectral envelopeestimation for later use by spectral regeneration decoding processes.

The receiver shown in FIG. 5 is similar to the receiver shown in FIG. 2and includes the estimator 30, which incorporates various aspects of thepresent invention to provide measures of magnitude and phase along thepaths 38 and 39, respectively. The estimator 30 may also provide MDSTcoefficients along the path 37. The decoder 26 uses these measures toobtain spectral components from encoded information received from thedeformatter 23. Examples of processes that may be used in the decoder26, which may depend on the measures of magnitude or phase, includeperceptual models used to determine adaptive quantization levels,spectral component synthesis from composite or coupled representations,and spectral component regeneration.

Devices that incorporate various aspects of the present invention may beimplemented in a variety of ways including software for execution by acomputer or some other apparatus that includes more specializedcomponents such as digital signal processor (DSP) circuitry coupled tocomponents similar to those found in a general-purpose computer. FIG. 9is a schematic block diagram of device 70 that may be used to implementaspects of the present invention. DSP 72 provides computing resources.RAM 73 is system random access memory (RAM) used by DSP 72 for signalprocessing. ROM 74 represents some form of persistent storage such asread only memory (ROM) for storing programs needed to operate device 70and to carry out various aspects of the present invention. I/O control75 represents interface circuitry to receive and transmit signals by wayof communication channels 76, 77. Analog-to-digital converters anddigital-to-analog converters may be included in I/O control 75 asdesired to receive and/or transmit analog signals. In the embodimentshown, all major system components connect to bus 71, which mayrepresent more than one physical bus; however, a bus architecture is notrequired to implement the present invention.

In embodiments implemented in a general purpose computer system,additional components may be included for interfacing to devices such asa keyboard or mouse and a display, and for controlling a storage devicehaving a storage medium such as magnetic tape or disk, or an opticalmedium. The storage medium may be used to record programs ofinstructions for operating systems, utilities and applications, and mayinclude embodiments of programs that implement various aspects of thepresent invention.

The functions required to practice various aspects of the presentinvention can be performed by components that are implemented in a widevariety of ways including discrete logic components, integratedcircuits, one or more ASICs and/or program-controlled processors. Themanner in which these components are implemented is not important to thepresent invention.

Software implementations of the present invention may be conveyed by avariety of machine readable media such as baseband or modulatedcommunication paths throughout the spectrum including from supersonic toultraviolet frequencies, or storage media that convey information usingessentially any recording technology including magnetic tape, cards ordisk, optical cards or disc, and detectable markings on media likepaper.

1. A method of processing information representing a source signalconveying content intended for human perception, the method comprising:receiving first spectral components that were generated by applicationof an analysis filterbank to the source signal, wherein the firstspectral components represent spectral content of the source signalexpressed in a first subspace of a multidimensional space; deriving oneor more first intermediate components from at least some of the firstspectral components, wherein at least some of the first intermediatecomponents differ from the first spectral components from which they arederived; forming a combination of the one or more first intermediatecomponents according to at least a portion of one or more impulseresponses to obtain one or more second intermediate components; derivingone or more second spectral components from the one or more secondintermediate components, wherein the second spectral componentsrepresent spectral content of the source signal expressed in a secondsubspace of the multidimensional space that includes a portion of themultidimensional space not included in the first subspace; obtainingestimated measures of magnitude or phase using the first spectralcomponents and the second spectral components; and applying an adaptiveprocess to the first spectral components to generate processedinformationgenerating encoded audio information using the first spectralcomponents, wherein the adaptive processgenerating step is responsive tothe estimated measures of magnitude or phase.; wherein the method isperformed by one or more computing devices.
 2. The method of claim 1,wherein: the first spectral components are transform coefficientsarranged in one or more blocks of transform coefficients that weregenerated by application of one or more transforms to one or moresegments of the source signal; and the portions of the one or moreimpulse responses are based on frequency response characteristics of theone or more transforms.
 3. The method of claim 2, wherein the frequencyresponse characteristics of the one or more transforms are dependent oncharacteristics of one or more analysis window functions that wereapplied with the one or more transforms to the one or more segments ofthe source signal.
 4. The method of claim 3, wherein at least some ofthe one or more transforms implement an analysis filter bank thatgenerates the first spectral components with time-domain aliasing. 5.The method of claim 3, wherein at least some of the one or moretransforms generate first spectral components having real valuesexpressed in the first subspace, and wherein the second spectral valueshave imaginary values expressed in the second subspace.
 6. The method ofclaim 5, wherein the transforms that generate first spectral componentshaving real values expressed in the first subspace are Discrete CosineTransforms or Modified Discrete Cosine Transforms.
 7. The method ofclaim 1, wherein: the first spectral components are transformcoefficients arranged in one or more blocks of transform coefficientsthat were generated by application of one or more transforms to one ormore segments of the source signal, the one or more second intermediatecomponents are obtained by combining the one or more first intermediatecomponents according to a portion of the one or more impulse responses,each of the one or more impulse responses comprise a respective set ofelements arranged in order, and the portion of each of the one or moreimpulse responses excludes every other element in the respective set ofelements.
 8. The method according to claim 1 that further comprisesobtaining estimated measures of magnitude or phase using one or morethird spectral components that are derived from at least some of the oneor more first spectral components.
 9. The method according to claim 8,wherein: the first spectral components are transform coefficientsarranged in one or more blocks of transform coefficients that weregenerated by application of one or more transforms to one or moresegments of the source signal; the third spectral components are derivedfrom a combination of two or more of the first spectral components; andthe estimated measures of magnitude or phase for a respective segment ofthe source signal are obtained adaptively using either the thirdspectral components or using the first and second spectral components.10. The method according to claim 8, wherein: the first spectralcomponents are transform coefficients arranged in one or more blocks oftransform coefficients that were generated by application of one or moretransforms to one or more segments of the source signal; the thirdspectral components are derived from a combination of two or more of thefirst spectral components; and the estimated measures of magnitude orphase for at least some spectral content of a respective segment of thesource signal are obtained using the third spectral components and theestimated measures of magnitude or phase for at least some of thespectral content of the respective segment of the source signal areobtained using the first and second spectral components.
 11. The methodaccording to claim 8 or 10 that comprises obtaining measures ofmagnitude or phase adaptively using either the third spectral componentsor using the first and second spectral components.
 12. The method ofclaim 1 that comprises adapting the portion of the one or more impulseresponses in response to a measure of spectral component significance.13. The method of claim 12, wherein the measure of spectral componentsignificance is provided by a perceptual model that assesses perceptualsignificance of the spectral content of the source signal.
 14. Themethod of claim 12, wherein the measure of spectral componentsignificance reflects isolation in frequency of one or more spectralcomponents.
 15. The method of claim 1, wherein: the first spectralcomponents are first transform coefficients arranged in one or moreblocks that were generated by application of one or more transforms toone or more segments of the source signal, a respective block having afirst number of first transform coefficients; the second spectralcomponents are second transform coefficients; a second number of secondtransform coefficients are derived that represent spectral content thatis also represented by some of the first transform coefficients in therespective block; and the second number is less than the first number.16. The method according to claim 1, 2, 9, 10 or 12 that comprises:applying the adaptive process to the first spectral components togenerate synthesized spectral components; deriving one or more thirdintermediate components from the first spectral components and/or thesecond spectral components and from the synthesized spectral components;and generating one or more output signals conveying content intended forhuman perception by applying one or more synthesis filterbanks to theone or more third intermediate components.
 17. The method according toclaim 16, wherein at least some of the synthesized spectral componentsare generated by spectral component regeneration.
 18. The methodaccording to claim 16, wherein at least some of the synthesized spectralcomponents are generated by decomposition of first spectral componentsand/or second spectral components representing a composite of spectralcontent for a plurality of source signals.
 19. The method according toclaim 16, wherein at least some of the synthesized spectral componentsare generated by combining first spectral components and/or secondspectral components to provide a composite representation of spectralcontent for a plurality of source signals.
 20. The method according toclaim 1, 2, 9, 10 or 12 that comprises: generating the first spectralcomponents by applying the analysis filter bank to the source signal;applying the adaptive process to the first spectral component togenerate encoded information representing at least some of the firstspectral components; and generating an output signal conveying theencoded information.
 21. A non-transitory computer readable mediumconveying a program of instructions that is executable by a device toperform a method of processing information representing a source signalconveying content intended for human perception, the method comprising:receiving first spectral components that were generated by applicationof an analysis filterbank to the source signal, wherein the firstspectral components represent spectral content of the source signalexpressed in a first subspace of a multidimensional space; deriving oneor more first intermediate components from at least some of the firstspectral components, wherein at least some of the first intermediatecomponents differ from the first spectral components from which they arederived; forming a combination of the one or more first intermediatecomponents according to at least a portion of one or more impulseresponses to obtain one or more second intermediate components; derivingone or more second spectral components from the one or more secondintermediate components, wherein the second spectral componentsrepresent spectral content of the source signal expressed in a secondsubspace of the multidimensional space that includes a portion of themultidimensional space not included in the first subspace; obtainingestimated measures of magnitude or phase using the first spectralcomponents and the second spectral components; and applying an adaptiveprocess to the first spectral components to generate processedinformationgenerating encoded audio information using the first spectralcomponents, wherein the adaptive processgenerating step is responsive tothe estimated measures of magnitude or phase.
 22. The medium of claim21, wherein: the first spectral components are transform coefficientsarranged in one or more blocks of transform coefficients that weregenerated by application of one or more transforms to one or moresegments of the source signal; and the portions of the one or moreimpulse responses are based on frequency response characteristics of theone or more transforms, which are dependent on characteristics of one ormore analysis window functions that were applied with the one or moretransforms to the one or more segments of the source signal.
 23. Themedium according to claim 21, wherein the method further comprisesobtaining estimated measures of magnitude or phase using one or morethird spectral components that are derived from at least some of the oneor more first spectral components.
 24. The medium according to claim 23,wherein: the first spectral components are transform coefficientsarranged in one or more blocks of transform coefficients that weregenerated by application of one or more transforms to one or moresegments of the source signal; the third spectral components are derivedfrom a combination of two or more of the first spectral components; andthe estimated measures of magnitude or phase for a respective segment ofthe source signal are obtained adaptively using either the thirdspectral components or using the first and second spectral components.25. The medium according to claim 23, wherein: the first spectralcomponents are transform coefficients arranged in one or more blocks oftransform coefficients that were generated by application of one or moretransforms to one or more segments of the source signal; the thirdspectral components are derived from a combination of two or more of thefirst spectral components; and the estimated measures of magnitude orphase for at least some spectral content of a respective segment of thesource signal are obtained using the third spectral components and theestimated measures of magnitude or phase for at least some of thespectral content of the respective segment of the source signal areobtained using the first and second spectral components.
 26. The mediumaccording to claim 23, wherein the method comprises obtaining measuresof magnitude or phase adaptively using either the third spectralcomponents or using the first and second spectral components.
 27. Themedium of claim 21, wherein the method comprises adapting the portion ofthe one or more impulse responses in response to a measure of spectralcomponent significance.
 28. The medium of claim 27, wherein the measureof spectral component significance is provided by a perceptual modelthat assesses perceptual significance of the spectral content of thesource signal.
 29. The medium of claim 27, wherein the measure ofspectral component significance reflects isolation in frequency of oneor more spectral components.
 30. The medium of claim 21, wherein: thefirst spectral components are first transform coefficients arranged inone or more blocks that were generated by application of one or moretransforms to one or more segments of the source signal, a respectiveblock having a first number of first transform coefficients; the secondspectral components are second transform coefficients; a second numberof second transform coefficients are derived that represent spectralcontent that is also represented by some of the first transformcoefficients in the respective block; and the second number is less thanthe first number.
 31. The medium according to claim 21, wherein themethod comprises: applying the adaptive process to the first spectralcomponents to generate synthesized spectral components; deriving one ormore third intermediate components from the first spectral componentsand/or the second spectral components and from the synthesized spectralcomponents; and generating one or more output signals conveying contentintended for human perception by applying one or more synthesisfilterbanks to the one or more third intermediate components.
 32. Themedium according to claim 21, wherein the method comprises: generatingthe first spectral components by applying the analysis filter bank tothe source signal; applying the adaptive process to the first spectralcomponent to generate encoded information representing at least some ofthe first spectral components; and generating an output signal conveyingthe encoded information.
 33. An apparatus for processing informationrepresenting a source signal conveying content intended for humanperception, the apparatus comprising: means for receiving first spectralcomponents that were generated by application of an analysis filterbankto the source signal, wherein the first spectral components representspectral content of the source signal expressed in a first subspace of amultidimensional space; means for deriving one or more firstintermediate components from at least some of the first spectralcomponents, wherein at least some of the first intermediate componentsdiffer from the first spectral components from which they are derived;means for forming a combination of the one or more first intermediatecomponents according to at least a portion of one or more impulseresponses to obtain one or more second intermediate components; meansfor deriving one or more second spectral components from the one or moresecond intermediate components, wherein the second spectral componentsrepresent spectral content of the source signal expressed in a secondsubspace of the multidimensional space that includes a portion of themultidimensional space not included in the first subspace; means forobtaining estimated measures of magnitude or phase using the firstspectral components and the second spectral components; and means forapplying an adaptive process to the first spectral components togenerate processed information generating encoded audio informationusing the first spectral components, wherein the adaptive processgenerating means is responsive to the estimated measures of magnitude orphase.
 34. The apparatus of claim 33, wherein: the first spectralcomponents are transform coefficients arranged in one or more blocks oftransform coefficients that were generated by application of one or moretransforms to one or more segments of the source signal; and theportions of the one or more impulse responses are based on frequencyresponse characteristics of the one or more transforms, which aredependent on characteristics of one or more analysis window functionsthat were applied with the one or more transforms to the one or moresegments of the source signal.
 35. The apparatus according to claim 33that further comprises means for obtaining estimated measures ofmagnitude or phase using one or more third spectral components that arederived from at least some of the one or more first spectral components.36. The apparatus according to claim 35 wherein: the first spectralcomponents are transform coefficients arranged in one or more blocks oftransform coefficients that were generated by application of one or moretransforms to one or more segments of the source signal; the thirdspectral components are derived from a combination of two or more of thefirst spectral components; and the estimated measures of magnitude orphase for a respective segment of the source signal are obtainedadaptively using either the third spectral components or using the firstand second spectral components.
 37. The apparatus according to claim 35,wherein: the first spectral components are transform coefficientsarranged in one or more blocks of transform coefficients that weregenerated by application of one or more transforms to one or moresegments of the source signal; the third spectral components are derivedfrom a combination of two or more of the first spectral components; andthe estimated measures of magnitude or phase for at least some spectralcontent of a respective segment of the source signal are obtained usingthe third spectral components and the estimated measures of magnitude orphase for at least some of the spectral content of the respectivesegment of the source signal are obtained using the first and secondspectral components.
 38. The apparatus according to claim 35 thatcomprises means for obtaining measures of magnitude or phase adaptivelyusing either the third spectral components or using the first and secondspectral components.
 39. The apparatus of claim 33 that comprises meansfor adapting the portion of the one or more impulse responses inresponse to a measure of spectral component significance.
 40. Theapparatus of claim 39, wherein the measure of spectral componentsignificance is provided by a perceptual model that assesses perceptualsignificance of the spectral content of the source signal.
 41. Theapparatus of claim 39, wherein the measure of spectral componentsignificance reflects isolation in frequency of one or more spectralcomponents.
 42. The apparatus of claim 33, wherein: the first spectralcomponents are first transform coefficients arranged in one or moreblocks that were generated by application of one or more transforms toone or more segments of the source signal, a respective block having afirst number of first transform coefficients; the second spectralcomponents are second transform coefficients; a second number of secondtransform coefficients are derived that represent spectral content thatis also represented by some of the first transform coefficients in therespective block; and the second number is less than the first number.43. The apparatus according to claim 33 that comprises: means forapplying the adaptive process to the first spectral components togenerate synthesized spectral components; means for deriving one or morethird intermediate components from the first spectral components and/orthe second spectral components and from the synthesized spectralcomponents; and means for generating one or more output signalsconveying content intended for human perception by applying one or moresynthesis filterbanks to the one or more third intermediate components.44. The apparatus according to claim 33 that comprises: means forgenerating the first spectral components by applying the analysis filterbank to the source signal; means for applying the adaptive process tothe first spectral component to generate encoded informationrepresenting at least some of the first spectral components; and meansfor generating an output signal conveying the encoded information.
 45. Amethod of processing information representing a source signal conveyingcontent intended for human perception, the method comprising: receivingfirst spectral components that were generated by application of a firstanalysis filterbank to the source signal; obtaining second spectralcomponents from a convolution of a derivation of at least some of thefirst spectral components in two or more segments with a portion of oneor more impulse responses of hypothetical filters related to thecombined effect of a synthesis filterbank inverse to the first analysisfilterbank and a second analysis filterbank coupled to the synthesisfilterbank; obtaining estimated measures of magnitude or phase using thefirst spectral components and the second spectral components; andgenerating encoded audio information using the first spectralcomponents, wherein the generating step is responsive to the estimatedmeasures of magnitude or phase; wherein the method is performed by oneor more computing devices.
 46. The method of claim 45, wherein: thefirst spectral components are transform coefficients arranged in one ormore blocks of transform coefficients that were generated by applicationof one or more transforms to one or more segments of the source signal;and the portions of the one or more impulse responses are based onfrequency response characteristics of the one or more transforms. 47.The method of claim 46, wherein the frequency response characteristicsof the one or more transforms are dependent on characteristics of one ormore analysis window functions that were applied with the one or moretransforms to the one or more segments of the source signal.
 48. Themethod of claim 47, wherein at least some of the one or more transformsimplement an analysis filter bank that generates the first spectralcomponents with time-domain aliasing.
 49. The method of claim 47,wherein at least some of the one or more transforms generate firstspectral components having real values and wherein the second spectralvalues have imaginary values.
 50. The method of claim 49, wherein thetransforms that generate first spectral components having real valuesare Discrete Cosine Transforms or Modified Discrete Cosine Transforms.51. The method according to claim 45 that further comprises obtainingestimated measures of magnitude or phase using one or more thirdspectral components that are derived from at least some of the one ormore first spectral components.
 52. The method according to claim 51,wherein: the first spectral components are transform coefficientsarranged in one or more blocks of transform coefficients that weregenerated by application of one or more transforms to one or moresegments of the source signal; the third spectral components are derivedfrom a combination of two or more of the first spectral components; andthe estimated measures of magnitude or phase for a respective segment ofthe source signal are obtained adaptively using either the thirdspectral components or using the first and second spectral components.53. The method according to claim 51, wherein: the first spectralcomponents are transform coefficients arranged in one or more blocks oftransform coefficients that were generated by application of one or moretransforms to one or more segments of the source signal; the thirdspectral components are derived from combination of two or more of thefirst spectral components; and the estimated measures of magnitude orphase for at least some spectral content of a respective segment of thesource signal are obtained using the third spectral components and theestimated measures of magnitude or phase for at least some of thespectral content of the respective segment of the source signal obtainedusing the first and second spectral components.
 54. The method accordingto claim 51 or 53 that comprises obtaining measures of magnitude orphase adaptively using either the third spectral components or using thefirst and second spectral components.
 55. The method of claim 45 thatcomprises adapting the portion of the one or more impulse responses inresponse to a measure of spectral component significance.
 56. The methodof claim 45, wherein the measure of spectral component significance isprovided by a perceptual model that assesses perceptual significance ofthe spectral content of the source signal.
 57. The method of claim 45,wherein the measure of spectral component significance reflectsisolation in frequency of one or more spectral components.
 58. Themethod of claim 45, wherein: the first spectral components are firsttransform coefficients arranged in one or more blocks that weregenerated by application of one or more transforms to one or moresegments of the source signal, a respective block having a first numberof first transform coefficients; the second spectral components aresecond transform coefficients; a second number of second transformcoefficients are derived that represent spectral content that is alsorepresented by some of the first transform coefficients in therespective block; and the second number is less than the first number.59. The method according to claim 45, 46, 52, 53 or 55 that comprisesapplying the adaptive process to the first spectral components togenerate synthesized spectral components.
 60. The method according toclaim 59, wherein at least some of the synthesized spectral componentsare generated by spectral component regeneration.
 61. The methodaccording to claim 59, wherein at least some of the synthesized spectralcomponents are generated by decomposition of first spectral componentsand/or second spectral components representing a composite of spectralcontent for a plurality of source signals.
 62. The method according toclaim 59, wherein at least some of the synthesized spectral componentsare generated by combining first spectral components and/or secondspectral components to provide a composite representation of spectralcontent for a plurality of source signals.
 63. The method according toclaim 45, 46, 52, 53 or 55 that comprises: generating the first spectralcomponents by applying the analysis filterbank to the source signal;applying the adaptive process to the first spectral component togenerate encoded information representing at least some of firstspectral components; and generating an output signal conveying theencoded information.
 64. A method of processing information representinga source signal conveying content intended for human perception, themethod comprising: receiving first spectral components that weregenerated by application of a first analysis filterbank to the sourcesignal; obtaining second spectral components from a convolution of aderivation of at least some of the first spectral components with aportion of one or more impulse responses of hypothetical filters relatedto the combined effect of a synthesis filterbank inverse to the firstanalysis filterbank and a second analysis filterbank coupled to thesynthesis filterbank; obtaining estimated measures of magnitude or phaseusing the first spectral components and the second spectral components;and generating encoded audio information using the first spectralcomponents, wherein the generating step is responsive to the estimatedmeasures of magnitude or phase; wherein the method is performed by oneor more computing devices.
 65. The method of claim 64, wherein: thefirst spectral components are transform coefficients arranged in one ormore blocks of transform coefficients that were generated by applicationof one or more transforms to one or more segments of the source signal;and the portions of the one or more impulse responses are based onfrequency response characteristics of the one or more transforms. 66.The method of claim 65, wherein the frequency response characteristicsof the one or more transforms are dependent on characteristics of one ormore analysis window functions that were applied with the one or moretransforms to the one or more segments of the source signal.
 67. Themethod of claim 66, wherein at least some of the one or more transformsimplement an analysis filter bank that generates the first spectralcomponents with time-domain aliasing.
 68. The method of claim 66,wherein at least some of the one or more transforms generate firstspectral components having real values and wherein the second spectralvalues have imaginary values.
 69. The method of claim 68, wherein thetransforms that generate first spectral components having real valuesare Discrete Cosine Transforms or Modified Discrete Cosine Transforms.70. The method according to claim 64 that further comprises obtainingestimated measures of magnitude or phase using one or more thirdspectral components that are derived from at least some of the one ormore first spectral components.
 71. The method according to claim 70,wherein: the first spectral components are transform coefficientsarranged in one or more blocks of transform coefficients that weregenerated by application of one or more transforms to one or moresegments of the source signal; the third spectral components are derivedfrom a combination of two or more of the first spectral components; andthe estimated measures of magnitude or phase are obtained adaptivelyusing either the third spectral components or using the first and secondspectral components.
 72. The method according to claim 70 wherein: thefirst spectral components are transform coefficients arranged in one ormore blocks of transform coefficients that were generated by applicationof one or more transforms to one or more segments of the source signal;the third spectral components are derived from combination of two ormore of the first spectral components; and the estimated measures ofmagnitude or phase for at least some spectral content of the sourcesignal are obtained using the third spectral components and theestimated measures of magnitude or phase for at least some of thespectral content of the respective segment of the source signal obtainedusing the first and second spectral components.
 73. The method accordingto claim 70 or 72 that comprises obtaining measures of magnitude orphase adaptively using either the third spectral components or using thefirst and second spectral components.
 74. The method of claim 64 thatcomprises adapting the portion of the one or more impulse responses inresponse to a measure of spectral component significance.
 75. The methodof claim 64, wherein the measure of spectral component significance isprovided by a perceptual model that assesses perceptual significance ofthe spectral content of the source signal.
 76. The method of claim 64,wherein the measure of spectral component significance reflectsisolation in frequency of one or more spectral components.
 77. Themethod of claim 64, wherein: the first spectral components are firsttransform coefficients arranged in one or more blocks that weregenerated by application of one or more transforms of the source signal,a respective block having a first number of first transformcoefficients; the second spectral components are second transformcoefficients; a second number of second transform coefficients arederived that represent spectral content that is also represented by someof the first transform coefficients in the respective block; and thesecond number is less than the first number.
 78. The method according toclaim 64, 65, 71, 72 or 74 that comprises applying the adaptive processto the first spectral components to generate synthesized spectralcomponents.
 79. The method according to claim 78, wherein at least someof the synthesized spectral components are generated by spectralcomponent regeneration.
 80. The method according to claim 78, wherein atleast some of the synthesized spectral components are generated bydecomposition of first spectral components and/or second spectralcomponents representing a composite of spectral content for a pluralityof source signals.
 81. The method according to claim 78, wherein atleast some of the synthesized spectral components are generated bycombining first spectral components and/or second spectral components toprovide a composite representation of spectral content for a pluralityof source signals.
 82. The method according to claim 64, 65, 66, 72 or74 that comprises: generating the first spectral components by applyingthe analysis filterbank to the source signal; applying the adaptiveprocess to the first spectral component to generate encoded informationrepresenting at least some of first spectral components; and generatingan output signal conveying the encoded information.
 83. A method ofprocessing information representing a source signal conveying contentintended for human perception, the method comprising: receiving firstspectral components that were generated by application of a firstanalysis filterbank to the source signal; obtaining second spectralcomponents from a convolution of a derivation of at least some of thefirst spectral components with a portion of one or more impulseresponses of hypothetical filters related to the combined effect of asynthesis filterbank inverse to the first analysis filterbank and asecond analysis filterbank coupled to the synthesis filterbank;obtaining estimated parameters using the first spectral components andthe second spectral components; and generating encoded audio informationusing the first spectral components, wherein the generating step isresponsive to the estimated measures of magnitude or phase; wherein themethod is performed by one or more computing devices.
 84. The method ofclaim 83, wherein: the first spectral components are transformcoefficients arranged in one or more blocks of transform coefficientsthat were generated by application of one or more transforms to one ormore segments of the source signal; and the portions of the one or moreimpulse responses are based on frequency response characteristics of theone or more transforms.
 85. The method of claim 84, wherein thefrequency response characteristics of the one or more transforms aredependent on characteristics of one or more analysis window functionsthat were applied with the one or more transforms to the one or moresegments of the source signal.
 86. The method of claim 85, wherein atleast some of the one or more transforms implement an analysis filterbank that generates the first spectral components with time-domainaliasing.
 87. The method of claim 85, wherein at least some of the oneor more transforms generate first spectral components having real valuesand wherein the second spectral values have imaginary values.
 88. Themethod of claim 87, wherein the transforms that generate first spectralcomponents having real values are Discrete Cosine Transforms or ModifiedDiscrete Cosine Transforms.
 89. The method according to claim 85,wherein the estimated parameters are estimated measures of magnitude orphase, further comprising obtaining estimated measures of magnitude orphase using one or more third spectral components that are derived fromat least some of the one or more first spectral components.
 90. Themethod according to claim 89, wherein: the first spectral components aretransform coefficients arranged in one or more blocks of transformcoefficients that were generated by application of one or moretransforms to one or more segments of the source signal; the thirdspectral components are derived from a combination of two or more of thefirst spectral components; and the estimated measures of magnitude orphase are obtained adaptively using either the third spectral componentsor using the first and second spectral components.
 91. The methodaccording to claim 89, wherein: the first spectral components aretransform coefficients arranged in one or more blocks of transformcoefficients that were generated by application of one or moretransforms to one or more segments of the source signal; the thirdspectral components are derived from combination of two or more of thefirst spectral components; and the estimated measures of magnitude orphase for at least some spectral content of the source signal areobtained using the third spectral components and the estimated measuresof magnitude or phase for at least some of the spectral content of therespective segment of the source signal obtained using the first andsecond spectral components.
 92. The method according to claim 89 or 91that comprises obtaining measures of magnitude or phase adaptively usingeither the third spectral components or using the first and secondspectral components.
 93. The method of claim 83 that comprises adaptingthe portion of the one or more impulse responses in response to ameasure of spectral component significance.
 94. The method of claim 83,wherein the measure of spectral component significance is provided by aperceptual model that assesses perceptual significance of the spectralcontent of the source signal.
 95. The method of claim 83, wherein themeasure of spectral component significance reflects isolation infrequency of one or more spectral components.
 96. The method of claim83, wherein: the first spectral components are first transformcoefficients arranged in one or more blocks that were generated byapplication of one or more transforms of the source signal, a respectiveblock having a first number of first transform coefficients; the secondspectral components are second transform coefficients; a second numberof second transform coefficients are derived that represent spectralcontent that is also represented by some of the first transformcoefficients in the respective block; and the second number is less thanthe first number.
 97. The method according to claim 83, 84, 90, 91 or 93that comprises applying the adaptive process to the first spectralcomponents to generate synthesized spectral components.
 98. The methodaccording to claim 97, wherein at least some of the synthesized spectralcomponents are generated by spectral component regeneration.
 99. Themethod according to claim 97, wherein at least some of the synthesizedspectral components are generated by decomposition of first spectralcomponents and/or second spectral components representing a composite ofspectral content for a plurality of source signals.
 100. The methodaccording to claim 97, wherein at least some of the synthesized spectralcomponents are generated by combining first spectral components and/orsecond spectral components to provide a composite representation ofspectral content for a plurality of source signals.
 101. The methodaccording to claim 83, 84, 90, 91 or 93 that comprises: generating thefirst spectral components by applying the analysis filterbank to thesource signal; applying the adaptive process to the first spectralcomponent to generate encoded information representing at least some offirst spectral components; and generating an output signal conveying theencoded information.